EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS 2025, Vol. 18, Issue 4, Article Number 7112 ISSN 1307-5543 – ejpam.com Published by New York Business Global Approximating Riemann-Stieltjes Integral Using New Time and Cost Efficient Trapezoid-Type Quadrature Kashif Memon1, Muhammad Mujtaba Shaikh2, Sara Mahesar2, Kamran Malik3, Hijaz Ahmad4,5,6,∗, Berna Uzun4, Ilker Ozsahin4,7, Waleed Mohammed Abdelfattah8,9 1 Institute of Mathematics and Computer Science, University of Sindh, Jamshoro, Sindh, Pakistan 2 Department of Basic Sciences and Related Studies, Mehran University of Engineering and Technology, Jamshoro, Sindh, Pakistan 3 Department of Mathematics, Government College University, Hyderabad, Pakistan 4 Operational Research Center in Healthcare, Near East University, Nicosia/TRNC, 99138 Mersin 10, Turkey 5 Sustainability Competence Centre, Széchenyi István University, Egyetem tér 1, H-9026 Győr, Hungary 6 Department of Mathematics, College of Science, Korea University, 145 Anam-ro, Seongbuk-gu, Seoul 02841, South Korea 7 Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai, Tamilnadu, India 8 College of Engineering, University of Business and Technology, Jeddah 23435, Saudi Arabia 9 Department of Engineering Mathematics and Physics, Faculty of Engineering, Zagazig University, P.O. 44519, Egypt Abstract. Engineers and scientists adopt numerical integration to achieve an approximate solution for definite integrals that have no analytic solution. This research focuses on developing some new derivative- based quadrature schemes for numerically integrating the integral of Riemann-Stieltjes (Rs-integral) by proposing new schemes which are based on trapezoid-type quadrature. The process of undetermined co- efficients has been used for the derivation of proposed schemes. The theoretical derivation and numerical verification of orders of accuracy have been addressed in line with the degrees of precision, and a sufficient improvement has been demonstrated over the existing schemes. The theorems regarding single and mul- tiple use of the suggested schemes in a finite interval have been proved along with the theoretical results on residual terms, both locally and globally. All suggested schemes have been verified to reduce to corre- sponding variants for the Riemann integral in the case when integrator g(t) = t. Numerical experiments have been performed on all the discussed schemes with the help of MATLAB coding. The experimental results assure the smaller numerical errors by the proposed schemes in the comparison of existing schemes. The obtained results show the efficiency of proposed schemes in light of computational burden and CPU time (in seconds) with reference to the existing quadrature. The proposed work substantially advances the existing knowledge with an addition of derivative-based correction terms in the usual quadrature in a way that the consequent computational costs and execution times are also minimized. 2020 Mathematics Subject Classifications: 65D30, 65D32 Key Words and Phrases: Quadrature rule, Riemann-Stieltjes integral, trapezoidal rule, derivative- based schemes, midpoint methods, means ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v18i4.7112 Email addresses: hijaz.ahmad@neu.edu.tr (H. Ahmad) https://www.ejpam.com 1 Copyright: © 2025 The Author(s). (CC BY-NC 4.0) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 2 of 38 1. Introduction One of the oldest and most attractive concepts in numerical analysis is numerical in- tegration. When integrating f(x) analytically is difficult or impossible, or if the values of f(x) are defined in tabular form, then the numerical integration methods are applied. However, numerical integration gives only the approximate value of the integral with given parameters. Numerical integration generates better results when it is solved properly with appropriate methods. The term quadrature is used to describe numerical integration in one dimension. Quadrature formulas are usually based on polynomial interpolation. Mostly, the conventional Newton-Cotes quadrature formulae work well for definite integrals. Dif- ferent types of Newton-Cotes formulas are known that are based on the range of the limits a and b. The Newton-Cotes formulae of closed type, which include both endpoints, can be regarded as a bulk of essential formulas. The closed Newton-Cotes integration formulae include the Trapezoid rule as a specific case. The integrand function is fitted with nth degree polynomials at n + 1 control points to obtain this formula. Mostly, attention of research works in the field of numerical integration was focused on the Riemann integral. Similar progress may be achieved for the RS-integral. The evaluation of RS-integral, in which the integrand is f(x) and the integrator is another function of x, α(x), devel- ops a mainstream part in mathematics [1]. RS-integral may be applied in the areas of statistics and probability theory of operators, mathematical analysis, etc. Obviously, the RS-integral can be regarded as an enhancement of the usual integration by Riemann, in a way that besides the integrand f , the integrator also depends on x. While a considerable effort in literature has been done on enhancing quadrature rules for the classical Riemann integrals as compared to the RS-integrals, only some investigations have been attempted to approximate the RS-integral. In 2008, Mercer [2] offered the Trapezoidal rule extension which addressed the applicability of the Hadamard integral inequality for the RS-integral. Moreover, in 2012, Mercer [3] introduced a new approach for midpoint and Simpson’s schemes on the RS-integral, based on the concept of relative convexity. In 2013, Zhao and Li [4] developed a novel midpoint derivative-based family of closed Newton-Cotes quadra- ture rules. Afterwards in 2014, Zhao et al. [5] proposed a trapezoid rule for the RS-integral based on the midpoint derivative, in which the function derivative was computed using the midpoint. In 2016, Shaikh et al. [6] improved with the second-order derivative-based rule with the help of midpoint, the Zhao and Li’s [4] scheme of Simpson’s 3/8-type, which had used fourth-order derivative. There are so many works that have been done on the numerical enhancement of the usual integration formulas of Newton and Cotes. In 2016, Ramachandran et al. [7] developed a novel quadrature of the closed form that used the geometric mean value to compute the derivative. They proved that this novel quadrature scheme provided an improvement of a single order in precision against the traditional closed quadrature by Newton and Cotes. In 2016, Ramachandran et al. [8] developed another closed-form novel quadrature approach that used the harmonic mean value to compute the derivative. They proved that this novel quadrature scheme provided an improvement of a single order in precision over the closed Newton-Cotes quadrature formula. Ramachandran et al. [9] performed a test K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 3 of 38 among three derivative-based closed Newton-Cotes quadrature schemes which were pro- posed in [4, 7, 8]. According to the comparison, the arithmetic mean derivative-based rule was much more effective than the other schemes. Moreover, Ramachandran et al. [10]-[11] developed the closed Newton-Cotes scheme to evaluate function derivatives using heronian mean and centroidal mean. They proved that these novel quadrature schemes provide an improvement of a single order in precision over the closed Newton-Cotes quadrature for- mula. The work in [12] encouraged proposition of time and cost effective quadrature schemes by utilizing derivative corrections as a better and efficient way to enhance the accuracy and precision of Riemann integral approximations as well. In recent years, some work has been focused on the approximation of RS-integral using different means, but with more nodal points in the sense of Simpson’s quadrature, for example [13], [14] and [15]. The applications of the RS-integral can be further studied through [16], whereas the studies [17] and [18] provide a way forward to apply the numerical procedures in the mesh-free sense to better deal with some case studies in engineering science. This research extends some recent derivative-based quadrature schemes of the Rie- mann integral into corresponding efficient derivative-based quadrature schemes for the RS-integral using different means. Further, the suggested schemes include the work of Zhao et al. [5] for the RS-integral with modification. The suggested schemes are much more effective than the Zhao et al. [5] scheme in comparison. The composite form is in- volved for the purpose of reducing errors. The derivations of suggested schemes are proved in composite form for the RS-integral and used for the numerical experiments. Finally, the theoretical and numerical results tailor the time and cost effectiveness of the proposed quadrature in terms of standard parameters, like: absolute error, cost, and CPU times. 2. Materials and Methods 2.1. Basics of Riemann and Riemann-Stieltjes integrals and their existing quadrature For integrating a function f(x) from x = a to x = b, the basic Riemann notation is: I(f ; a, b) = ∫ b a f(x) dx (1) Geometrically, I(f ; a, b)represents area bounded between f(x) and X-axis when x varies from a to b. The RS-integral advances the Riemann concept of integration, which was limited to integrating a function f(x) in a segment of the set of real numbers. The RS- integral utilizes integrator which is another function of x, and also monotonic rising but bounded. Therefore, it is important to frame and analyze the latter instance geometrically in three dimensions. The RS-integral notation can be describes as: RS(f ; g; a, b) = ∫ b a f(t) dα(t) (2) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 4 of 38 Figure 1: Geometrical representation of Riemann integral where integrator is increasing there with both continuous integrand and integrator in the interval of integration. The width of a sub-rectangle can be computed by differences in the precise values of x in that sub-interval, which are the interval’s end points in the Riemann integral. However, the similar partitioning is done using the differences in the integrator increasing function’s values in the RS-integral. The geometrical form of the Riemann integral is shown in Fig. 1 , where the x-axis is used to divide the area into rectangles, and limiting case of areas summed-up define the integral. The similar explanation on the RS-integral is explained in Fig.2. The graphs of integrator and integrand can be seen initially in 2D plane, each versus the X-axis, and then the rectangular regions that cover the surface area using the RS-integral are identified on the three-dimensional graph. Figure 2: Geometrical representation of Riemann-Stieltjes integral [19] The Riemann notion of integral I(f ; a, b) through (1), when evaluated numerically using a quadrature scheme, is the sum of products of the integrand evaluated at some finite discrete points and the corresponding weights. We can write with reference to [1]: I(f ; a, b) = ∫ b a f(x) dx ≈ n∑ i=0 wif(xi) (3) The quantities wi and xi = a + ih are, respectively, the weights and abscissas/nodes, where h = b−a n and the index i varies from 0 to n. The nodes can be the equally-spaced points in the interval of integration – as the case of widely used quadrature by Newton and Cotes – or also zeros of orthogonal polynomials in the sense of quadrature introduced by Gauss. The most basic techniques are the Newton-Cotes quadrature formulas. The closed Newton-Cotes quadrature formulas include the end points of integration interval [a, b] for evaluating the integral that is described as: I(f ; a, b) = ∫ b a f(x) dx ≈ n∑ i=0 wif(x0 + ih) (4) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 5 of 38 There are several methods for calculating the weighting coefficients in equation (4). One basic approach of a quadrature formula is concerned with the precision, in which the weight values w0, w1, , wnare considered to have a zero approximation error in equation (4), i.e. E(f) = ∫ b a f(x) dx− n∑ i=0 wif(xi) = 0 (5) where f(x) are monic polynomials , where degree j varies from 0 to n. A few important metrics related to accuracy and errors associated to quadrature rules are explained here for accommodating better understanding of the forthcoming sections. Definition 1. If a quadrature formula is able to exactly integrate all polynomials of degree up to n, but not onward, then we say that it has a degree of exactness/precision of n. Definition 2. Order of accuracy of a quadrature formula refers to the exponent of the truncation error term, and it indicates decrease of error as step-size decreases. Definition 3. General truncation error in a quadrature rule which has degree of exactness of p can be expressed as: R[f ] = Ch(p+1)f (p+1)(ϵ) (6) where c is a nonzero constant, h is the width of interval and p + 1is the order of error term with precision p. Listed below are a few recognized members of the closed-form quadrature by Newton and Cotes for approximating Riemann integration [1] with local approximations and truncation errors:∫ b a f(x) dx = b− a 2 [f(a) + f(b)]− (b− a)3 12 f ′′(ξ) (7) ∫ b a f(x) dx = b− a 6 [ f(a) + 4f ( a+ b 2 ) + f(b) ] − (b− a)5 2880 f (4)(ξ) (8)∫ b a f(x) dx = b− a 8 [ f(a) + 3f ( 2a+ b 3 ) + 3f ( a+ 2b 3 ) + f(b) ] − (b− a)5 6480 f (4)(ξ) (9) whereξ ∈ (a, b). The precision of the Trapezoidal rule is 1, and its order of accuracy is 2, as it gives an exact proper integral of polynomials of degree one or less. The precision of the Simpson’s 3-point (1/3) rule is 3, and its order of accuracy is 4, as it gives an exact proper integral of polynomials of degree three or less. The rule (9) has same precision 3 as of (8), and order of accuracy 4. Besides the usual closed-form rules by Newton and Cotes, there exist quadrature proposed in last decade which use derivatives and correction terms in the usual closed-from rules [4], [7], [8], [10], [11]. Here, we refer to the local approximations and truncation error terms of such quadrature. The arithmetic mean derivative-based Trapezoidal rule [4] has exactness degree of 3, and is described as:∫ b a f(t) dt = b− a 2 [f(a) + f(b)]− (b− a)3 12 f ′′ ( a+ b 2 ) − (b− a)5 480 f (4)(ξ), (10) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 6 of 38 The geometric mean derivative-based Trapezoidal rule [7] with degree of exactness 2 is described as:∫ b a f(t) dt = b− a 2 [f(a) + f(b)]− (b− a)3 12 f ′′ (√ ab ) − (b− a)3 24 (√ b− √ a )2 f (3)(ξ), (11) The harmonic mean derivative-based Trapezoidal rule [8] with degree of exactness 2 is described as: ∫ b a f(t) dt = b− a 2 [f(a) + f(b)]− (b− a)3 12 f ′′ ( 2ab a+ b ) − (b− a)5 24(a+ b) f (4)(ξ), (12) The heronian mean derivative-based Trapezoidal rule [10] with degree of exactness 2 is described as:∫ b a f(t) dt = b− a 2 [f(a) + f(b)]−(b− a)3 12 f ′′ ( a+ √ ab+ b 3 ) −(b− a)3 72 (√ b− √ a )2 f (3)(ξ), (13) The centroidal mean derivative-based Trapezoidal rule [11] with degree of exactness 2 is described as: ∫ b a f(t) dt = b− a 2 [f(a) + f(b)]− (b− a)3 12 f ′′ ( 2 ( a2 + ab+ b2 ) 3(a+ b) ) + (b− a)5 72(a+ b) f (4)(ξ), (14) One can note that the formula (10) can be said, comparatively in contrast with others (11)-(14), to have no limitations on the choice of limits of integration. On the other hand, methods (11)-(14), are prone to limitations based on the limits of integration. For example, (11) and (13) lead to complex number arithmetic if limits of integration are opposite in sign. In this way, (12) and (14) lead to indeterminate and undefined forms when limits of integration add up to zero. It should also be noted that the * in (12) and (14) refers to the studies [8] and [11], respectively, where authors put fourth-order derivative in the error term, whereas it is expected that an interpolatory quadrature has (p + 2)nd order derivative in the local error term if its precision is p. The extension of the conventional Trapezoid rule of the Riemann integral is discussed in [20] for the approximation of RS-integral. So, its resulting conventional Trapezoid rule without derivative for RS integral with local error term is as follows: RS(f ; g;α, β) = ∫ β α f(x) dg = T +RT [f ] = ( 1 β − α ∫ β α g(t) dt− g(α) ) f(α) + ( g(β)− 1 β − α ∫ β α g(t)dt ) f(β)− (β − α)3 12 f ′′(ξ)g′(η), (15) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 7 of 38 where ξ, η ∈ (α, β) The rule (15) describes the formula of two points, so its performance of large integration intervals is not appropriate. In order to get smaller local errors in each approximation, the composite scheme of original Trapezoid rule, referred here as CT scheme with global error term, for RS integrals given in [20] is: ∫ β α f(t) dg = CT +RCT [f ] = [ η β − α ∫ x1 α g(t) dt− g(α) ] f(α) + η β − α n−1∑ k=1 [∫ xk+1 xk g(t)dt− ∫ xk xk−1 g(t)dt ] f(xk) + [ g(β)− η β − α ∫ β xn−1 g(t)dt ] f(β)− (β − α)3 12n2 f ′′(µ)g′′(η), (16) where, µ, η ∈ (α, β) Zhao et al. [21] discussed the conventional Trapezoidal rule with the midpoint deriva- tive for the RS integral. So, its resulting scheme is as follows:∫ β a f(x) dg ≈ ZT = T + (∫ β a ∫ t a g(x) dx dt− β − a 2 ∫ β a g(t) dt ) f ′′(c), (17) where, c = (−2β2 + α2 − αβ) ∫ β α g(t) dt+ 6β ∫ β α ∫ t α g(x) dx dt− 6 ∫ β α ∫ t α ∫ y α g(x) dx dy dt 6 ∫ β α ∫ t α g(x) dx dt− 3(β − α) ∫ β α g(t) dt , (18) This method has precision 3 and its error term R[f] is as follows: α3 + αβ2 + α2β2 − 3β3 + 6(β − α)c2 24 [∫ β α g(t) dt+ β2 − α2 2 (∫ β α ∫ t α g(x) dx dt ) − β ∫ β α ∫ t α ∫ y α g(x) dx dt+ ∫ β α ∫ t α ∫ z α ∫ y α g(x) dx dy dz dtf (4)(ξ), (19) where ξ ∈ (α, β) While in [5] Zhao et al. attempted to modify the existing trapezoid- type scheme without derivatives by introducing derivative of second-order at point c, the improvement happens to be quite complicated and demands a lot of computational effort due to complicated formula of c in the complete formula. Further, in scheme (17), when g(t) = t, then expression of c reduces to (α+β)3 2(α−β)2 instead of (α+β) 2 . Consequently, the scheme of Zhao et al. [5] was rectified by modified Zhao Trapezoidal rule (MZT) in [22]. The MZT scheme is: K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 8 of 38∫ β a f(x) dg ≈ MZT = ( 1 β − a ∫ β α g(t) dt− g(a) ) f(a)+ ( g(β)− 1 β − a ∫ β a g(t) dt ) f(β) + (∫ β a ∫ t a g(x) dx dt− β − a 2 ∫ β a g(t) dt ) f ′′(c), (20) where c = (−2β2 + α2 + αβ) ∫ β α g(t) dt+ 6β ∫ β α ∫ t α g(x) dx dt− 6 ∫ β α ∫ t α ∫ y α g(x) dx dy dt 6 ∫ β α ∫ t α g(x) dx dt− 3(β − α) ∫ β α g(t) dt , (21) This MZT method has precision 3 and its error term R[f] is as follows: α2 + αβ2 + α2β − 3β3 + 6(β − α)c2 24 (∫ β α g(t) dt+ β2 − c2 2 [∫ β α ∫ t α g(x) dx dt − β ∫ β α ∫ t α ∫ y α g(x) dx dy dt+ ∫ β α ∫ t α ∫ x α ∫ y α g(x) dx dy dz dtf∗(4)(ξ), (22) where, ξ ∈ (α, β) 2.2. Proposed Trapezoid-type quadrature for approximating Riemann- Stieltjes integral The main motivation in the present study lies on proposing time and cost efficient trapezoid-type quadrature basing the contributions around the simplicity of c in the for- mula introduced in [5]. Because of the complicated c in (17), its dependence on the inte- grand and integrator functions, its dependence on derivatives of integrator and integrals of integrator, it is imperative to think of precise and simple cin (17). In this attempt, we con- sider the analogies from recent advancements in the case of Riemann integrals where the derivatives were used in addition to the actual trapezoid approximations to enhance the accuracy [4], [7], [8], [10], [11]. The points at which derivatives where used in these studies were statistical averages of the limits of the interval of integration. Selecting such means in the proposed trapezoid-type quadrature for the RS-integral to evaluate derivatives in advance minimizes the computational cost and execution times, without compromising on the accuracy of the quadrature. It will be demonstrated that our enhancements prove to be efficient than those suggested and used on this subject through [5] and [22]. The contributions are divided in two sections here: firstly we derive and prove theorems on the proposed trapezoid-type quadrature using arithmetic mean, and secondly the similar discussion in precisely extended to other averages. 2.2.1. Approximating Riemann-Stieltjes integral using arithmetic mean trapezoid- type quadrature In the formula (17), we restrict c to be the mid-point of the limits of integration, or in other words, arithmetic average of the limits, then consequent trapezoid-type approximation so obtained is referred as AMT, and the quadrature is derived in Theorem 1. K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 9 of 38 Theorem 1. Assuming continuity of f(t) and g(t) in along with g(t) being monotonically rising in the same interval, AMT quadrature for the RS-integral is:∫ β α f(t) dg ≈ AMT = ( 1 β − α I1 − g(α) ) f(α) + ( g(β)− 1 β − α I1 ) f(β) + ( I2 − β − α 2 I1 ) f ′ ( α+ β 2 ) , (23) where I1 = ∫ β α g(t) dt, I2 = ∫ β α ∫ t α g(x) dx dt, I3 = ∫ β α ∫ t α ∫ y α g() dx dy dt, and I4 = ∫ β α ∫ t α ∫ z α ∫ y α g(x) dx dy dz dt. Proof. Referencing to the AMT scheme for the Riemann integral as in [4], the proposed AMT quadrature for the RS-integral can be described generally as:∫ β α f(t) dg ≈ a0f(α) + b0f(β) + c0f ′ ( α+ β 2 ) . (24) Looking at the precision of (1) [4], we can expect the degree of exactness of (24) to be 3 as well. AMT quadrature in (24) when integrand is allowed to be a monomial of degree at most 3 leads to four equations:∫ β α 1dg = a0 + b0,∫ β α tdg = a0α+ b0β,∫ β α t2dg = a0α 2 + b0β 2 + 2c0,∫ β α t3dg = a0α 3 + b0β 3 + 3c0(α+ β). The following system of equations (25)-(29) has been obtained using the integration tech- nique by parts of the RS integral. a0 + b0 = g(α) + (β), (25) a0α+ b0β = α β−αI1 − αg(α) + βg(β)− β β−αI1 ⇒ a0α+ b0β = βg(β)− αg(α)− I1 (26) aα2 + bβ2 + 2c0 = ( α2 β − α − β2 β − α ) I1 − αz2g(α) + β2g(β) + 2I1 − (β − α)2I1 (27) ⇒ aα2 + b0β 2 + 2c0 = β2g(β)− 2βI + 2I2 (28) a0α 3+b20β 3+3c0(α+β) = α3 β−αI1−α3g(α)−β3g(β)− β3 β−αI1+3(α+β)I2− 3 2(β 2−α2)I1 ⇒ a0α 3 + b0β 3 + 3c0(α+ β) = β3g(β)− α3g(α)− 3β2I1 + 6βI2 − 6I3 (29) The following matrix is obtained from the system of linear equations (25)-(29) in the un- knowns: K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 10 of 38 M =  1 1 0 α β 0 α2 β2 2 α3 β3 3(α+ β)  The above matrix M is further written in reduced row echelon form as: M ≈ MR =  1 0 0 0 1 0 0 0 1 0 0 0  It is observed fromMR that the rank(M) = 3, since the reduced row echelon form has three nonzero rows. So, we solve the above equations (25), (26), and (28) simultaneously to determine the coefficients a0, b0andc0. a0 = 1 β−αI1g(α) b0 = g(β)− 1 β−αI1 c0 = (β−α 2 )I1 + I2 Putting the values of coefficients a0, b0andc0 in (24), we have:∫ β α f(t) dg ≈ [ 1 β − α I1 − g(α)]f(α) + [g(β)− 1 β − α I1]f(β) + [( β − α 2 )I1 + I2]f ′ ( α+ β 2 ) . (30) This formula is the required AMT quadrature, as in (23). Theorem 2 presents truncation error in the AMT quadrature locally. Theorem 2. Assuming continuity of f(t) and g(t) in along with g(t) being monotonically rising in the same interval [α, β], AMT quadrature’s truncation error for the RS-integral is: RAMT[f ] = ( 2α3 + 2αβ2 − 6β3 + 3(β − α)(α+ β)2 48 I1 ) f (4)(ξ)g(η) + ( 4β2 − (α+ β)2 8 I2 − I2 βI3 + I4 ) f (4)(ξ)g(η) (31) Proof. Taking for the truncation error, as degree of exactness of (23) is 3, we have: RAMT[f ] = 1 4! ∫ β α t4 dg −AMT (t4; g;α, β) (32) From [4], we know that: 1 4! ∫ β α t4 dg = 1 24 ((β4g(β))− α4g(α))− β3 6 I1 − βI3 + I4 (33) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 11 of 38 By Theorem 1 and scheme (23), we have: AMT (t4; g;α, β) = α2 4! ( 1 β − α I1 − g(α) ) + β2 4! ( g(β)− 1 β − α I1 ) + (α+ β)2 8 ( I2 − β − α 2 I1 ) 1 2! ( α+ β 2 ) 2 (34) RAMT [f ] = 1 24 [(β4g(β)−α4g(α))− (β)3 6 I1+ (β)2 2 I2−βI3+ I4+ α4 24(β − α) I1+ α4g(α) 24 − β4g(β) 24 + β4 24(β − α) I1 − (α+ β)2 8 I2 + (β − α)(α+ β)2 16 ]f (4)(ξ)g(η) (35) RAMT [f ] = ( 2α3 + 2αβ2 + 2α2β − 6β3 + 3(β − α)(α+ β)2 48 ) I1 + [ 4β2 − (α− β)2 8 I2 − βJ3 + I4 ] f (4)(ξ)g(η) (36) Theorem 3. If g(t) = t, then AMT quadrature and its truncation error term as in (23)-(24) for RS integral reduces to simple AMT quadrature (10) [4]. Proof. Obtaining the following using the Theorem 2:∫ β a f(t) dg = ∫ β a f(t) dt = ( 1 β − α ∫ β a t dt− g(α) ) f(α) + ( g(β)− 1 β − α ∫ β a t dt ) f(β) + (∫ β a ∫ t a x dx dt− β − α 2 ∫ β a t dt ) f ′′ ( α+ β 2 ) + 2α3 + 2αβ2 + 2α2β − 6β3 + 3(β − α)(α+ β)2 48 ∫ β a t dt + 4β2 − (α+ β)2 8 (∫ β a ∫ t a x dx dt− β ∫ β a ∫ t a ∫ y a x dx dy dz ) f (4)(ξ)g′(η) + (∫ β a ∫ t a ∫ z a ∫ y a x dx dy dz dt ) f (4)(ξ)g′(η) (37) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 12 of 38 Now, it is easy to obtain:∫ β a t dt = β2−α2 2∫ β a ∫ t a x dx dt = β3 6 − α2β 2 + α3 3∫ β a ∫ t a ∫ y a x dx dy dt = β4 24 − α2β2 4 + α3β 3 − α4 8 . and, using these in (37), we finally get:∫ β a f(t) dt = β − α 2 [f(α) + f(β)]− (β − α)3 12 f ′′ ( α+ β 2 ) − (β − α)5 480 f (4)(ξ), (38) where which show the reducibility of proposed AMT quadrature its simple counterpart (10) [4]. It is a fact that the composite quadrature rules provides greater accuracy than the basic formulae. To reduce truncation error further from local to global application of AMT quadrature, the composite quadrature to (23) is referred here as AMCT, and with b−a n = h keeping the assumptions of Theorems 1-2 it can be written as:∫ β a f(t) dg ≈ AMCT = [ n β − α ∫ x1 a g(t) dt− g(α) ] f(α) + n β − α n−1∑ k=1 [∫ xk+1 xk g(t) dt− ∫ xk xk−1 g(t) dxi ] f(xk) + [ g(β)− n β − α ∫ β xn−1 g(t) dt ] f(β) + n−1∑ i=1 [∫ xk xk−1 ∫ t xk−1 g(x) dx dt− h 2 ∫ k xk−1 g(t) dt ] f ′′ ( xk−1 + xk 2 ) (39) Theorem 4. Assuming continuity of f(t) and g(t) in β−α 2 along with g(t) being monotoni- cally rising in the same interval. Suppose that [α.β] partitioned in n sub-intervals [xk, xk−1] of size h = β−α 2 in form of uniformly-spaced integration points: , for k = 0, 1, , n. The AMCT quadrature with global truncation error RAMCT [f ] is: ∫ β a f(t) dg = AMCT +RAMCT [f ] = [ n β − α ∫ x1 a g(t) dt− g(α) ] f(α) + n β − α n−1∑ k=1 [∫ xk+1 xk g(t) dt− ∫ xk xk−1 g(t) dxi ] f(xk) + [ g(β)− n β − α ∫ β xn−1 g(t) dt ] f(β) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 13 of 38 + n−1∑ k=1 [∫ xk xk−1 ∫ xt xk−1 g(x) dx dt− h 2 ∫ xk xk−1 g(t) dt ] f ′′ ( xi+1 + xi 2 ) + n [ 2α3 + 2αβ2 + 2α2β − 6β3 + 3(β − α)(α+ β)2 48 ∫ β a g(t) dt + 4β2 − (α+ β)2 8 ∫ β a ∫ t a g(x) dx dt −β ∫ β a ∫ t a ∫ y a g(x) dx dy dt+ ∫ β a ∫ t a ∫ y a ∫ z a g(x) dx dy dz dt ] f (4)(ξ)g′(η) (40) whereξ, η ∈ (α, β) Proof. Refereeing to a local truncation error from (24) for the AMT quadrature, we have:[ 2x3p−1 + 2xp−1x 2 p + 2x2p−1xp − 6x3p + 3(xp − xp−1)(xp−1 + xp) 2 48 ∫ xp xp−1 g(t) dt + 4x2p − (xp−1 + xp) 2 8 ∫ xp xp−1 ∫ t xp−1 g(x) dx dt− xp ∫ xp xp−1 ∫ t xp−1 ∫ y xp−1 g(x) dx dy dt + ∫ xp xp−1 ∫ t xp−1 ∫ y xp−1 ∫ z xp−1 g(x) dx dy dz dt ] f (4)(ξp) g ′(ηp) (41) where ξp, η ∈ (α, β) Adding n such local truncation errors leads to the global truncation error as: n−1∑ k=1 [ 2x3k−1 + 2xk−1x 2 k + 2x2k−1xk − 6x3k + 3(xk − xk−1)(xk−1 + xk) 2 48 ∫ xk xk−1 g(t) dt + 4x2k − (xk−1 + xk) 2 8 ∫ xk xk−1 ∫ t xk−1 g(x) dx dt− xk ∫ xk xk−1 ∫ t xk−1 ∫ y xk−1 g(x) dx dy dt + ∫ xk xk−1 ∫ t xk−1 ∫ y xk−1 ∫ z xk−1 g(x) dx dy dz dt ] f (4)(ξk) g ′(η) (42) = n ( 2α3 + 2αβ2 + 2α2β − 6β3 + 3(β − α)(α+ β)2 48 ∫ β a g(t) dt + 4β2 − (α+ β)2 8 ∫ β a ∫ β a g(x) dx dt − β ∫ β a ∫ t a ∫ y a g(x) dx dy dt + ∫ β a ∫ t a ∫ y a ∫ z a g(x) dx dy dz dt )[ 1 n n∑ k=1 f (4)(ξk) g ′(η) ] (43) Let M = [ 1 n ∑n k=1 f (4)(ξk) g ′(η) ] K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 14 of 38 Clearly,minx∈[α,β] { f (4)(x)g(4)(x) } ≤ M ≤ maxx∈[α,β] { f (4)(x)g(4)(x) } . Because f (4)(x)and g′(x) in [α, β] are continuous, so there is a point µ such thatM = f (4)(µ)g′(η) . Finally, the global truncation error RAMCT [f ] takes the form: = n ( 2α3 + 2αβ2 + 2α2β − 6β3 + 3(β − α)(α+ β)2 48 ∫ β a g(t) dt+ 4β2 − (α+ β)2 8 ∫ β a ∫ β a g(x) dx dt − β ∫ β a ∫ t a ∫ y a g(x) dx dy dt+ ∫ β a ∫ t a ∫ y a ∫ z a g(x) dx dy dz dt ) f (4)(µ)g′(η) (44) where µ, η ∈ (α, β) and h = β−α n 2.2.2. Proposed derivative-based rules for the Riemann-Stieltjes integral using other means Some other schemes are derived in terms of geometric, harmonic, heronian and centroidal means for RS-integral using the approach discussed in section 2.2.1. The derivation of the quadrature scheme, its error term and the consequent reduction to the classical Riemann integral case are in a same pattern as proved in Theorems 1-4, respectively. For brevity, we discuss the main results in form of corollaries here that have similar proofs to Theorems 1-4, and can also be considered immediate consequences of the process followed therein. In scheme (17), selecting the expression of c as the geometric, harmonic, heronian and centroidal means of the limits of integration, then the corresponding new trapezoid-type schemes for the RS integrals, labeled as: GMT, HaMT, HeMT and CMT, respectively, are described in Corollaries 1-4 with error terms in basic form. Corollary 1. Assuming continuity of f(t) and g(t) in [α, β] along withg(t) being mono- tonically rising in the same interval, GMT quadrature for the RS-integral is:∫ β a f(x) dg ≈ GMT = ( 1 β − α ∫ β a g(t) dt− g(α) ) f(α) + ( g(β)− 1 β − α ∫ β a g(t) dt ) f(β) + (∫ β a ∫ t a g(x) dx dt− β − α 2 ∫ β a g(t) dt ) f ′′ (√ αβ ) (45) with precision 2. The error term RGMT [f ] is:( α2 + αβ − 2β2 + 3 √ αβ(β − α) 6 ∫ β α g(t) dt+ ( β − √ αβ )∫ β α ∫ t α g(x) dx dt − ∫ β α ∫ t α ∫ y α g(x) dx dy dt ) f (3)(ξ), K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 15 of 38 where ξ ∈ (α, β) Corollary 2. Assuming continuity of f(t) and g(t) in [α, β] along with g(t) being mono- tonically rising in the same interval, HaMT quadrature for the RS-integral is:∫ β a f(x) dg ≈ HaMT = ( 1 β − α ∫ β a g(t) dt− g(α) ) f(α) + ( g(β)− 1 β − α ∫ β a g(t) dt ) f(β) + (∫ β a ∫ t a g(x) dx dt− β − α 2 ∫ β a g(t) dt ) f ′′ ( 2αβ α+ β ) (46) with precision 2. The truncation error RHaMT [f ]is:( α3 − 4α2β + 5αβ2 − 2β3 6(α+ β) ∫ β α g(t) dt+ ( β2 − αβ α+ β )∫ β α ∫ t α g(x) dx dt − ∫ β α ∫ t α ∫ y α g(x) dx dy dt ) f (4)(ξ), (47) where ξ ∈ (α, β) Corollary 3. Assuming continuity of f(t) and g(t) in [α, β] along with g(t) being mono- tonically rising in the same interval, HeMT quadrature for the RS-integral is:∫ β a f(x) dg ≈ HeMT = ( 1 β − α ∫ β a g(t) dt− g(α) ) f(α) + ( g(β)− 1 β − α ∫ β a g(t) dt ) f(β) + (∫ β a ∫ t a g(x) dx dt− β − α 2 ∫ β a g(t) dt ) f ′′ ( α+ √ αβ + β 3 ) (48) with precision 2. The error term RHeMT [f ] is( (β − α)( √ αβ − β) 6 ∫ β a g(t) dt+ 2β − α− √ αβ 3 ∫ β a ∫ t a g(x) dx dt − ∫ β a ∫ t a ∫ y a g(x) dx dy dt ) f (3)(ξ), (49) where ξ ∈ (α, β) Corollary 4. Assuming continuity of f(t)andg(t) in [α, β] along with g(t) being mono- tonically rising in the same interval, CMT quadrature for the RS-integral is:∫ β a f(x) dg ≈ CMT = ( 1 β − α ∫ β a g(t) dt− g(α) ) f(α) + ( g(β)− 1 β − α ∫ β a g(t) dt ) f(β) + (∫ β a ∫ t a g(x) dx dt− β − α 2 ∫ β a g(t) dt ) f ′′ ( 2(α2 + αβ + β2) 3(α+ β) ) (50) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 16 of 38 with precision 2. The error term RCMT [f ] is:( 2α2β − αβ2 − α3 6(α+ β) ∫ β α g(t) dt+ αβ + β2 − 2α2 3(α+ β) ∫ β α ∫ t α g(x) dx dt − ∫ β α ∫ t α ∫ y α g(x) dx dy dt ) f (3)(ξ), (51) where ξ ∈ (α, β) It is trivial to verify the reduction of the GMT, HaMT, HeMT and CMT rules for the RS integral to the classical Riemann integral by choosing g(t) = t in context of the proof of Theorem 3 for the AMT rule. Using g(t) = t in (44), (46), (48) and (50) along with in the error terms (1), (47), (49) and (51), we obtain the corresponding schemes (11)-(14) for the Riemann integral [7]. The successful reducibility of proposed derivative- based rules to the counterparts for Riemann integral show that the proposed rules defined in Theorems 1-5 and Corollaries 1-4 are a sensible modification to the existing rules, the classical Trapezoid and Zhao et al. [5] rules. To guarantee higher accuracy and lower errors, the global/composite quadrature through (44), (46), (48) and (50) are referred as GMCT, HaMCT, HeMCT and CMCT, and are presented in Corollary 5. Corollary 5. Assuming continuity of f(t) and g(t)in [α, β] along with g(t) being mono- tonically rising in the same interval, and with, the composite form of the proposed schemes (45), (46), (48) and (50) are given in (52)-(55).∫ β a f(t) dg ≈ GMCT = [ n β − α ∫ x1 a g(t) dt− g(α) ] f(α) + n β − α n−1∑ k=1 [∫ xk+1 xk g(t) dt− ∫ xk xk−1 g(t) dt ] f(xk) + [ g(β)− n β − α ∫ β xn−1 g(t) dt ] f(β) + n∑ k=1 [∫ xk xk−1 ∫ t xk−1 g(x) dx dt− h 2 ∫ xk xk−1 g(t) dt ] f ′′(√xkxk−1 ) (52) ∫ β a f(t) dg ≈ HaMCT = [ n β − α ∫ x1 a g(t) dt− g(α) ] f(α) + n β − α n−1∑ k=1 [∫ xk+1 xk g(t) dt− ∫ xk xk−1 g(t) dt ] f(xk) + [ g(β)− n β − α ∫ β xn−1 g(t) dt ] f(β) + n∑ k=1 [∫ xk xk−1 ∫ t xk−1 g(x) dx dt− h 2 ∫ xk xk−1 g(t) dt ] f ′′ ( 2xkxk−1 xk−1 + xk ) (53) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 17 of 38 ∫ β a f(t) dg ≈ HeMCT = [ n β − α ∫ x1 a g(t) dt− g(α) ] f(α) + n β − α n−1∑ i=k [∫ xk+1 xk g(t) dt− ∫ xk xk−1 g(t) dt ] f(xk) + [ g(β)− n β − α ∫ β xn−1 g(t) dt ] f(β) + n∑ k=1 [∫ xk xk−1 ∫ t xk−1 g(x) dx dt− h 2 ∫ xi xk−1 g(t) dt ] f ′′ ( xk−1 + √ xk−1xi + xk 3 ) (54) ∫ β a f(t) dg ≈ CMCT = [ n β − α ∫ x1 a g(t) dt− g(α) ] f(α) + n β − α n−1∑ k=1 [∫ xk+1 xk g(t) dt− ∫ xk xk−1 g(t) dt ] f(xk) + [ g(β)− n β − α ∫ β xn−1 g(t) dt ] f(β) + n∑ k=1 [∫ xk xk−1 ∫ t xk−1 g(x) dx dt− h 2 ∫ xi xk−1 g(t) dt ] f ′′ ( 2 ( x2k−1 + xk−1xk + x2k ) 3 (xk−1 + xk) ) (55) Unlike the standard form of the truncation error (43) of the proposed AMCT quadra- ture, the same cannot be expected for the case of proposed trapezoid-type quadrature with other averages. This is because of the non-conventional expressions of type ( √ (b)− √ (a)), (α+ β), etc. in local truncation errors in addition to the usual conventional exponents of (β − α); as also obvious from the definition 3. However, the ZCT [5] quadrature’s composite form, though not presented in [5], takes the form: K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 18 of 38 ∫ β a f(t) dg ≈ ZCT = [ n β − α ∫ x1 a g(t) dt− g(α) ] f(α) + n β − α n−1∑ k=1 [∫ xk+1 xk g(t) dt− ∫ xk xk−1 g(t) dt ] f(xk) + [ g(β)− n β − α ∫ β xn−1 g(t) dt ] f(β) + n∑ k=1 [∫ xk xk−1 ∫ t xk−1 g(x) dx dt− h 2 ∫ xk xk−1 g(t) dt ] g(t)f ′′(εk) (56) n ( α3 + αβ2 + α2β − 3β3 + 6(β − α)c2 24 ∫ β α g(t) dt+ β2 − c2 2 ∫ β α ∫ t α g(x) dx dt − β ∫ β α ∫ t α ∫ y α g(x) dx dy dt+ ∫ β α ∫ t α ∫ y α ∫ z α g(x) dx dy dz dt ) f (4)(ξ) g′(η) (57) where µ, η ∈ [α, β]. Now, we describe the composite form MZCT of MZT [22] in (57). ∫ β a f(t) dg ≈ MZCT = [ n β − α ∫ x1 a g(t) dt− g(α) ] f(α) + n β − α n−1∑ k=1 [∫ xk+1 xk g(t) dt− ∫ xk xk−1 g(t) dt ] f(xi) + [ g(β)− n β − α ∫ β xn−1 g(t) dt ] f(β) + n∑ k=1 [∫ xk xk−1 ∫ t xk−1 g(x) dx dt− h 2 ∫ xk xk−1 g(t) dt ] g(t)f ′′(εi) (58) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 19 of 38 n ( α3 + αβ2 + α2β − 3β3 + 6(β − α)c2 24 ∫ β α g(t) dt+ β2 − c2 2 ∫ β α ∫ t α g(x) dx dt − β ∫ β α ∫ t α ∫ y α g(x) dx dy dt+ ∫ β α ∫ t α ∫ y α ∫ z α g(x) dx dy dz dt ) f (4)(ξ) g′(η) (59) where µ, η ∈ (α, β). 3. Results of the numerical experiments and their discussion We examine the performance of the proposed quadrature of Trapezoid-type for RS- integral over original Trapezoid, Zhao et al. [5] and modified Zhao-Trapezoid (MZT) [22] rules in terms of absolute error distributions when integration strips are increased. In ad- dition, we also compute computational/observed order of accuracy/exactness of proposed quadrature so that the theoretical expressions in Theorems 1-4 and Corollaries 1-5 are verified. It should be noted that in the previous studies [3], [5], the numerical experiments were not performed on the quadrature schemes for the RS-integral ; however, this research verifies the theoretical results by experimental results solving five different examples. Five problems are taken from [21], [23],[24], [13] and [14], and these are tested in approx- imations up to 16 decimal places of accuracy using the MATLAB software. The results were noted using the Intel (R) Core (TM) Laptop using 4GB RAM operating at speed of 0.80 GHz – 1.00 GHz. Also, the double precision default arithmetic of MATLAB is utilized for the results. (i) Example 1 ∫ 3.5 4.5 sin 5x d(cosx) = 0.227676016130689 (ii) Example 2 ∫ 6 5 sinx d(x3) = 59.655908136641912 (iii) Example 3 ∫ 6 5 ex d(sinx) = 187.4269314248657 (iv) Example 4 ∫ pi 0 x3 d(sinx) = −17.608813203268074 (v) Example 5 ∫ 1 −1 x 3 d(ex) = −17.608813203268074 The formula of absolute error drop (AED) [9] is: AED = |EV AV | (60) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 20 of 38 In (60), AE and AV refer to exact and approximate values of integrals, respectively. The formula for the observed/computational order of accuracy is defined in [25], [24], [13], [14] as: p = ln ( |N(2h)−N(0)| |N(h)−N(0)| ) ln 2 . (61) where N indicates approximate quadrature result at the step-size indicated in the paren- thesis, and with 0 inside the base result – the exact integral value – is intended. Figs. (3)-(7), respectively, display AEDs for the case of examples 1-5 for the quadra- ture approximations of previous quadrature: Trapezoid CT [20], ZCT [5] and MZCT [22], and the new trapezoid-type quadrature: AMCT,GMCT,HMCT,HeMCT and CMCT. For numerical test integrals 1-3 through Figs.(3)-(5), it is obvious to note the ascending per- formance of the proposed quadrature. The AEDs for the CMCT quadrature are lowest of all for test numerical integrals 1-3, and the AEDs in the case of existing quadrature CT and ZCT are much slower (3)-(5). The ZCT quadrature [5] is less likely to assure the claimed units of precision and accuracy as in [5], but MZCT quadrature [22] exhibits obvious improvement over the ZCT quadrature in all test integrals. The ZCT quadrature behaves poorest of all in the case of problems 4-5. In problems 4-5 (see Figs. (6)-(7)), the new quadrature proposed in this study result in smaller AEDs when compared with ZCT and CT quadrature. Also, the MZCT quadrature [22] assures smallest AED than what was expected in single strip, but then the AEDs do not get smaller rapidly since we used double precision default in MATLAB. To avoid the distrubance in results due to ZCT oscillations in decreasing errors, the AEDS for examples 1-5 are also displayed in Figs. (8)-(12) without the ZCT errors. The Figs. (8)-(12) provide more comprehensive and clearer picture of the performance of the new proposed rules with each other and the CT in terms of decreasing AEDs. Table (1) summarizes quadrature results for problems 1-5 and corresponding AEDs at the final integration strip, which evidently confirms that the proposed quadrature assures smaller AEDs against ZCT and CT in all problems taken; the CMCT quadrature being the best in problems 1-3. The results of MZCT quadrature [22] are pretty inclined near the proposed quadrature for problems 1-3, but the default double precision limit in problems 4-5 restricts MZCT quadrature to rapidly reduce errors ahead. Hence, the proposed quadrature exhibit stable pattern of AEDs. In tables (2)-(6), we present the computational/observed results on the orders of exact- ness/accuracy for all used quadrature for the case of problems 1-5 through the methodology described in [22]. For the case of ZCT quadrature with reference to the tables (2)-(4), the numerical results on order of exactness are not in line with what was expected through [5]. However, the MZCT and all the proposed quadrature results tend to the expected theoretical results. The MZCT quadrature results on numerical order of accuracy cannot be actualized because of precision limits in the case of problems 4-5; ACT quadrature also did not result in expected accuracy in [5]. On the other hand, all the proposed quadrature actualize the theoretically expected results in terms of the numerical results presented in Tables (2)-(6), with the only exception of the two: GMCT and HeMCT for the problem 5 alone through Table 6. The exception is result of the fact that these two rules have K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 21 of 38 limitation for the limits of integration being different signs; this was the case in problem 5. Similarly, there is an exception for the two quadrature formulae: HMCT and CMCT for the problem 5 as zero denominators are caused since limits add to zero, but only for odd strips. Both the rules are applicable safely in the alternate cases with even strips. We highlight that the selection of numerical problems in this study was done in the form of problems 1-5 intentionally from the literature so that the best coverage of use, restrictions and performance of all the quadrature used can be done sufficiently. The quadrature formulae in the case compared previous and new rules differ in the fact that these utilize different information of integrand and integrator functions, their derivatives and integra- tions. Hence, it is mandatory to observe the computational overhead/cost through the information content used in the formulae and also discuss the CPU usage time (in sec- onds) when all the quadrature formulae are applied to solve problems 1-5 under similar conditions to achieve a given error bound. Table (7) presents the distribution of evalu- ations in the formulae of all used quadrature. Tables (8)-(12) summarize distribution of computational cost for all used quadrature when the preset error tolerance was kept at 10−5 for problems 1-5. Also, the net computational costs for problems 1-5 are displayed in Tables (8)-(12). The results in Tables (8)-(12) and corresponding last columns in these highlight the cost effectiveness of the proposed quadrature formulae over ZCT, CT and MZCT in all problems, but with exception for for GMCT and HeMCT quadrature for problem 5 only. From Tables (11)-(12) in the case of problems 4-5, it is again reconfirmed that the ZCT quadrature is not applicable; the MZCT quadrature restrict to first strip of integration only and all the new quadrature show better performance for the cost effi- ciency. In the same way to achieve a preset AED of at most 10−5, the performance of all the used quadrature was tested with regards to the time efficiency, and these results are presented in Table (13) for all problems in form of execution times (in seconds). Table 13 highlights that the AMCT and CMCT quadrature exhibit smallest times mostly, except for problems 4-5 than MZCT quadrature, which is restricted to frit strip only. Finally, it can be safely concluded that the proposed quadrature exhibit encouraging and better performance overall than the previous CT and ZCT quadrature. K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 22 of 38 Table 1: Absolute errors are compared to all schemes for Examples 1-5 Trapezoid Variant Example 1 (n = 50) Example 2 (n = 200) Example 3 (n = 50) Example 4 (n = 50) Example 5 (n = 50) CT 0.227486276308570 −59.655783853372625 187.4331788489458 −17.612760265633682 0.450095792679060 (1.8974× 10−4) (1.2428× 10−4) (6.2474× 10−3) (3.9471× 10−3) (5.8839× 10−4) ZCT 0.227524664135771 −59.655762104211938 187.4331788489458 −29.345389591343285 −1.30536248.6941216 (1.5135× 10−4) (1.4603× 10−4) (6.2474× 10−3) (1.1737× 10+1) (1.7549× 10+0) MZCT 0.227676063572853 −59.655908136719574 1.874269314873392 −17.608813203268074 0.449507401824980 (4.7442× 10−8) (7.7648× 10−11) (6.2473× 10−8) (1.9201× 10−15) (−7.1054× 10−15) AMCT 0.227676057801290 −59.655908136739335 187.4269315224149 −17.608813463050069 0.449507451964994 (4.1671× 10−8) (9.7423× 10−11) (9.7549× 10−8) (2.5978× 10−7) (5.0140× 10−8) GMCT 0.227676058321505 −59.655908136813444 187.4269315777802 −17.608807372830086 0.449086933331309 (4.2191× 10−8) (1.7153× 10−10) (1.5291× 10−7) (5.8304× 10−6) (4.2047× 10−4) HaMCT 0.227676058841074 −59.655908136887568 187.4269316331454 −17.608805228593440 0.449508128616925 (4.2710× 10−8) (2.4566× 10−10) (2.0828× 10−7) (7.9747× 10−6) (7.2679× 10−7) HeMCT 0.227676057974770 −59.655908136764033 187.4269315408704 −17.608811432976740 0.449367279087097 (4.1844× 10−8) (1.2212× 10−10) (1.1600× 10−7) (1.7703× 10−6) (1.4012× 10−4) CMCT 0.227676057454114 −59.655908136689931 187.4269314855043 −17.608816207868944 0.449507226414350 (4.1323× 10−8) (4.8018× 10−11) (6.0639× 10−8) (3.0046× 10−6) (1.7541× 10−7) K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 23 of 38 Figure 3: Distributions of absolute-error-drops in Example 1 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 24 of 38 Figure 4: Distributions of absolute-error-drops in Example 2 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 25 of 38 Figure 5: Distributions of absolute-error-drops in Example 3 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 26 of 38 Figure 6: Distributions of absolute-error-drops in Example 4 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 27 of 38 Figure 7: Distributions of absolute-error-drops in Example 5 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 28 of 38 Figure 8: Distributions of absolute-error-drops without ZCT in Example 1 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 29 of 38 Figure 9: Distributions of absolute-error-drops without ZCT in Example 2 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 30 of 38 Figure 10: Distributions of absolute-error-drops without ZCT in Example 3 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 31 of 38 Figure 11: Distributions of absolute-error-drops without ZCT in Example 4 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 32 of 38 Figure 12: Distributions of absolute-error-drops without ZCT in Example 5 Table 2: . Representation of the order of accuracy to all schemes for Example 1 m n CT ZCT MZCT AMCT GMCT HaMCT HeMCT CMCT 1 1 NA NA NA NA NA NA NA NA 2 2 2.5728 3.3788 5.0724 5.1766 4.9006 4.4721 5.0970 5.3064 4 4 2.0420 4.3106 4.1474 4.1447 4.1338 4.1124 4.1422 4.1459 8 8 2.0091 -0.8154 4.0348 4.0338 4.0317 4.0272 4.0334 4.0338 16 16 2.0022 2.1713 4.0086 4.0083 4.0078 4.0067 4.0082 4.0083 32 32 2.0006 3.2118 4.0021 4.0021 4.0020 4.0017 4.0021 4.0021 64 64 2.0001 1.1185 4.0006 4.0006 4.0005 4.0004 4.0005 4.0005 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 33 of 38 Table 3: Representation of the order of accuracy to all schemes for Example 2 m n CT ZCT MZCT AMCT GMCT HaMCT HeMCT CMCT 5 5 NA NA NA NA NA NA NA NA 10 10 2.0018 1.9243 4.0025 4.0022 4.0017 4.0010 4.0020 4.0030 20 20 2.0004 2.4487 4.0007 4.0006 4.0004 4.0003 4.0005 4.0008 40 40 2.0001 2.3985 4.0001 4.0002 4.0002 4.0000 4.0001 4.0002 80 80 2.0001 1.4026 4.0000 4.0000 4.0000 4.0001 4.0000 4.0001 160 160 2.0000 2.4557 3.9998 4.0001 4.0001 4.0001 4.0001 4.0002 Table 4: Representation of the order of accuracy to all schemes for Example 3 m n CT ZCT MZCT AMCT GMCT HaMCT HeMCT CMCT 1 1 NA NA NA NA NA NA NA NA 2 2 1.9319 1.9319 3.8984 3.9697 3.9467 3.9283 3.9610 3.9970 4 4 1.9844 1.9844 3.9761 3.9935 3.9878 3.9832 3.9913 4.0003 8 8 1.9962 1.9962 3.9941 3.9984 3.9971 3.9958 3.9979 4.0002 16 16 1.9990 1.9990 3.9985 3.9996 3.9992 3.9990 3.9995 4.0000 32 32 1.9998 1.9998 3.9996 3.9999 3.9998 3.9998 3.9999 4.0000 64 64 1.9999 1.9999 3.9999 4.0000 4.0000 3.9999 4.0000 4.0000 Table 5: Representation of the order of accuracy to all schemes for Example 4 m n CT ZCT MZCT AMCT GMCT HaMCT HeMCT CMCT 1 1 NA NA NA NA NA NA NA NA 2 2 0 NA NA 4.3044 1.5452 1.9113 3.6353 2.6795 4 4 1.8345 0.1972 NA 4.0662 3.2559 2.9653 3.0169 3.3438 8 8 1.9649 0.0404 NA 4.0160 3.6533 3.4400 3.5898 3.5701 16 16 1.9916 0.0095 NA 4.0040 3.7589 3.6076 3.7252 3.6758 32 32 1.9979 0.0025 NA 4.0010 3.8029 3.6944 3.7801 3.7374 64 64 1.9995 0.00061 NA 4.0002 3.8289 3.7487 3.8118 3.7784 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 34 of 38 Table 6: Representation of the order of accuracy to all schemes for Example 5 m n CT ZCT MZCT AMCT GMCT HaMCT HeMCT CMCT 1 1 NA NA NA NA NA NA NA NA 2 2 0 53.7424 1.0000 3.9032 3.9499 NA 3.7927 NA 4 4 1.7226 51.8971 1.0000 3.9747 3.3325 4.0089 3.6427 4.0217 8 8 1.9363 0.3302 1.5850 3.9936 1.0357 4.0023 0.9225 4.0055 16 16 1.9844 0.0725 NA 3.9984 1.8139 4.0005 1.8014 4.0014 32 32 1.9962 0.0176 NA 3.9996 1.9516 4.0001 1.9489 4.0003 64 64 1.9990 0.0044 NA 3.9999 1.9867 4.0000 1.9861 4.0001 Table 7: Total Computational cost in Trapezoidal variants for n sub-intervals Trapezoidal Variants f g f (1) f (2) I1 I2 I3 Total evaluations Total (n=1) CT n+ 1 2 0 0 n 0 0 2n+ 3 5 ZCT n+ 1 2 0 n n n n 5n+ 3 8 MZCT n+ 1 2 0 n n n n 5n+ 3 8 AMCT n+ 1 2 0 n n n 0 4n+ 3 7 GMCT n+ 1 2 0 n n n 0 4n+ 3 7 HaMCT n+ 1 2 0 n n n 0 4n+ 3 7 HeMCT n+ 1 2 0 n n n 0 4n+ 3 7 CMCT n+ 1 2 0 n n n 0 4n+ 3 7 Table 8: Computational cost in Trapezoidal variants to achieve ξ ≤ 10−5 in Example 1 Trapezoidal Variants f g f (1) f (2) I1 I2 I3 Total evaluations CT 219 2 0 0 218 0 0 439 ZCT 209 2 0 208 208 208 208 1043 MZCT 15 2 0 14 14 14 14 73 AMCT 14 2 0 13 13 13 0 55 GMCT 14 2 0 13 13 13 0 55 HaMCT 14 2 0 13 13 13 0 55 HeMCT 14 2 0 13 13 13 0 55 CMCT 14 2 0 13 13 13 0 55 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 35 of 38 Table 9: Computational cost in Trapezoidal variants to achieve ξ ≤ 10−5 in Example 2 Trapezoidal Variants f g f (1) f (2) I1 I2 I3 Total evaluations CT 707 2 0 0 706 0 0 1415 ZCT 701 2 0 700 700 700 700 3503 MZCT 16 2 0 15 15 15 15 78 AMCT 13 2 0 12 12 12 0 51 GMCT 14 2 0 13 13 13 0 55 HaMCT 16 2 0 15 15 15 0 63 HeMCT 13 2 0 12 12 12 0 51 CMCT 11 2 0 10 10 10 0 43 Table 10: Computational cost in Trapezoidal variants to achieve ξ ≤ 10−5accuracy in Example 3 Trapezoidal Variants f g f (1) f (2) I1 I2 I3 Total evaluations CT 1251 2 0 0 1250 0 0 2503 ZCT 1251 2 0 1250 1250 1250 1250 6253 MZCT 16 2 0 15 15 15 15 78 AMCT 17 2 0 16 16 16 0 67 GMCT 19 2 0 18 18 18 0 75 HaMCT 20 2 0 19 19 19 0 79 HeMCT 18 2 0 17 17 17 0 71 CMCT 15 2 0 14 14 14 0 59 Table 11: Computational cost in Trapezoidal variants to achieve ξ ≤ 10−5accuracy in Example 4 Trapezoidal Variants f g f (1) f (2) I1 I2 I3 Total evaluations CT 995 2 0 0 994 0 0 1991 ZCT – – – – – – – – MZCT – – – – – – – – AMCT 22 2 0 21 21 21 0 87 GMCT 45 2 0 44 44 44 0 179 HaMCT 49 2 0 48 48 48 0 195 HeMCT 33 2 0 32 32 32 0 131 CMCT 38 2 0 37 37 37 0 151 K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 36 of 38 Table 12: Computational cost in Trapezoidal variants to achieve ξ ≤ 10−5 accuracy in Example 5 Trapezoidal Variants f g f (1) f (2) I1 I2 I3 Total evaluations CT 285 2 0 0 284 0 0 571 ZCT – – – – – – – – MZCT – – – – – – – – AMCT 15 2 0 14 14 14 0 59 GMCT 327 2 0 326 326 326 0 1307 HaMCT 27 2 0 26 26 26 0 107 HeMCT 189 2 0 188 188 188 0 755 CMCT 21 2 0 20 20 20 0 83 Table 13: Average CPU time (in seconds) to achieve |ε| ≤ 10−5 in examples 1-5 Trapezoidal Variants Example 1 Example 2 Example 3 Example 4 Example 5 CT 68.043837 12.821123 432.303629 364.293037 39.670771 ZCT 552.603703 153.984100 6469.382491 – – MZCT 28.010080 5.261296 27.353811 – – AMCT 14.760520 3.085717 14.702231 20.346118 7.468816 GMCT 14.929914 3.146952 16.340413 42.374740 203.371200 HaMCT 14.880674 3.230150 17.171850 46.938636 11.446324 HeMCT 14.782009 3.061632 15.484834 30.986400 105.331456 CMCT 14.773490 2.944814 13.170858 35.370088 9.399820 4. Conclusion This study focused the development of new variants of trapezoid-type quadrature keep- ing track on the efficiency in terms of time and cost effectiveness for the RS-integral approximation. The derivative corrections have been utilized at the points which were averages of the limits of integration. The derivation of proposed rules was discussed along with proved theorems on the local and global applications, local and global truncation errors and reduction to conventional forms. An exhaustive numerical examination was conducted for the thorough performance evaluation of the previous and new quadrature by considering varying nature test RS- integrals from literature. The new quadrature enhance precision and accuracy of the previously existing quadrature. The numerical re- sults on error drops, accuracy (orders observed computationally), execution times and computational overheads in all considered problems reflected encouraging behavior and application of the proposed quadrature over existing ones. Besides, we also highlighted the critical cases indicating natural limitations and usability of the used quadrature for the RS-integrals. The proposed quadrature appear to be efficient enhancements, both in terms of accuracy and computational cost, over the existing ones. The contributions of K. Memon et al. / Eur. J. Pure Appl. Math, 18 (4) (2025), 7112 37 of 38 this study can be useful in future studies focusing the fractional integration and solving integral and integro-differential equations with different weight functions. References [1] R. L. Burden and J. D. Faires. Numerical Analysis. Brooks/Cole, Boston, MA, USA, 9th edition, 2011. [2] P. R. Mercer. Hadamard’s inequality and trapezoid rules for the riemann-stieltjes integral. 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