BIBECHANA Vol. 20, No. 2, August 2023, 126-133 ISSN 2091-0762 (Print), 2382-5340 (Online) Journal homepage: http://nepjol.info/index.php/BIBECHANA Publisher:Dept. of Phys., Mahendra Morang A. M. Campus (Tribhuvan University)Biratnagar Caputo-Fabrizio approach to numerical fractional derivatives Shankar Pariyar1,2, Jeevan Kafle2∗ 1Tri-Chandra Multiple Campus, Ghantaghar, Kathmandu, Nepal, 2Central Department of Mathematics, Tribhuvan university, Kathmandu, Nepal ∗Corresponding author.Jeevan Kafle,: jeevan.kafle@cdmath.tu.edu.np Abstract Fractional calculus is an essential tool in every area of science today. This work gives the quadratic interpolation-based L1-2 formula for the Caputo-Fabrizio derivative, a numerical technique for approximating the fractional derivative. To get quadratic and cubic convergence rates, respectively, we study the use of Lagrange interpolation in the L1 and L1-2 formula- tions. Our numerical analysis shows the accuracy of the theory’s predicted convergence rates. The L1-2 formula aims to enhance the accuracy and usability of a flexible tool for many ap- plications in science and mathematics. We demonstrate the validity of the theory’s predicted convergence rates using numerical analysis. Several numerical examples are also given to show how the suggested approaches may be utilized to determine the Caputo-Fabrizio deriva- tive of well-known functions. Lagrange interpolation is used in the L1 and L1-2 procedures to obtain quadratic and cubic convergence rates, respectively. The numerical study demonstrates that the L1-2 formula offers greater accuracy when compared to current approaches. In ad- dition, it is a better apparatus for several applications in science and mathematics. Due to its higher convergence rate, the L1-2 formula outperforms other available numerical methods for scientific computations. The L1-2 formula, a novel numerical method for the Caputo- Fabrizio derivative that makes use of quadratic interpolation, is introduced in this study as a conclusion. Keywords Caputo fractional derivatives, Numerical solution, Lagrange interpolation, L1 formula, L1-2 formula. Article information Manuscript received: April 7, 2023; Accepted: April 25, 2023 DOI https://doi.org/10.3126/bibechana.v20i2.53971 This work is licensed under the Creative Commons CC BY-NC License. https://creativecommons. org/licenses/by-nc/4.0/ 1 Introduction A branch of mathematics known as numerical anal- ysis for fractional calculus is concerned with the creation and use of numerical techniques for the so- lution of fractional calculus-related issues [1]. The study of fractional calculus is still in its infancy and is continually changing. Integrating and differenti- ating non-integer categories is the main concept be- hind it. It is extensively employed across a range of scientific and technological fields of study, including the sciences of physics, engineering, biology, eco- 126 http://nepjol.info/index.php/BIBECHANA jeevan.kafle@cdmath.tu.edu.np https://doi.org/10.3126/bibechana.v20i2.53971 https://creativecommons.org/licenses/by-nc/4.0/ https://creativecommons.org/licenses/by-nc/4.0/ Shankar Pariyar and Jeevan Kafle / BIBECHANA 20 (2023) 126-133 127 nomics, and others [1]. In contrast to fractional integral equations, which involve fractional integrals, fractional differential equations are defined by the presence of fractional derivatives. A system is said to have a differ- ential order if its behavior can be correctly pre- dicted by one of these equations, both of them, or both together [2]. Models based on fractional- order differential equations have been developed to describe complex dynamical systems and physical phenomena. To advance the theory and applica- tion of fractional order differentials, committed re- searchers have made substantial efforts. The frac- tional derivatives rule is not a collection of rules that can be applied in every situation, although the terms fractional differentiation and integration have been defined differently by various mathemati- cians [3]. Because Riemann-Liouville (R-L) shows that the derivative of a constant component is not zero, it has been demonstrated that it can be diffi- cult to calculate fractional derivatives using regular calculus. The R-L fractional derivative is the mean- ing of these that is most widely used. Constant terms have non-zero derivatives, as demonstrated by the R-L discovery, which demonstrates that the derivative of a constant term is not zero, In the framework of classical calculus, it is difficult to an- alyze fractional derivatives [4, 5]. Jumarie expertly overcame this difficulty in the most recent version by modernizing the idea of F.D. of the R-L category. The fractional deriva- tive versions of Caputo, Grunwald-Letinikov (G-L), and Jumarie’s modified R-L are the most success- ful in avoiding the problem of non-zero derivatives resulting from a constant function [2]. While the Grunwald-Letinikov (G-L) term is useful for nu- merical applications, When it comes to analytical techniques, the concepts of R-L and Caputo are especially useful. No one technique can be used to solve the linear fractional differential equations. The derivative of the Mittag-Leffler function (MLF) [6] can be calculated using the fractional derivative with the Jumarie modification. Swedish scientist ML developed the MLF for aver- aging divergent series at the beginning of the 20th century. Given its crucial part in supplying so- lutions with integral and derivatives of fractional order, MLF is a special transcendental function. The fractional order derivatives or integrations of the superdiffusively transport, random walk, kinetic equation, and complex systems are readily solved by the MLF [4,7]. Both the standard and extended MLF interpolate phenomena are governed by shared kinetic equa- tions and their fractional analogs between an expo- nential law and a power law [8, 9]. In contrast to linear fracti,onal differential equations, which can- not be solved by a single technique, the FD of the MLF function can be derived by using the updated definition of the FD proposed by Jumarie. The ex- ponential function ez is essential for integer-order differential equations, according to theory. G.M. Mittag-Leffler was the first to refer to the function that is now denoted by, with its one parameter gen- eralization [10], was cited by G.M. Mittag-Leffler as the source first [7]. Fractional calculus can be used to solve fractional differential equations and explain odd phenomena. It is closely linked to MLF and its expansions. The two-parameter MLF, R(β) > 0, was established by Goreflo et al and Agarwal [11] immediately after the introduction of this version with a second complex parameter. Leibniz asked L’ Hospital [12] about the conse- quences of n = 1 2 and the nth derivative of the f(t) = t linear model. FC was developed in re- sponse to this query. Leibniz discovered a paradox that had human consequences, but he also noted the creation of FC on September 30, 1695. A 34-year delay later, in 1849, he added the beta and gamma functions to it and referenced to G.H. 1849 [13]. La- grange [14] developed and published the exponent principle for integer-order differential operators in 1772. Some of the most significant definitions of FC from several eminent mathematicians are given by us. Numerous mathematicians, including Fourier, Lacroix, Riemann, Liouville, Caputo, Euler, and G- L [15], were inspired by Romero’s work and his cita- tion by Laplace [16] in 1812 to advance the subject of fractional calculus. They also made use of Leg- endre’s extended factorial notation and Lacroix’s simple nth-order derivative [11]. By using the Laplace transform approach, Hum- bert and Agarwal [15] created a number of connec- tions for this study. Solving oscillation and relax- ation equations is governed by first and second or- ders of linear differential equations [17,18]. Caputo and Fabrizio [19] in 2015 have introduced a new type of fractional derivative called CFD to remove the singular kernel. Using both Caputo-Fabrizio and Atangana-Baleanu derivatives, Algahtani [20] (2016) carried out a numerical comparison of the Allen-Cahn equation Gao et al. [13] in (2014) intro- duced a novel fractional numerical technique, the L1-2 formula, for the Caputo fractional derivative, which improved the accuracy of numerical solutions from δt2−α to δt3−α via quadratic interpolation. The aforementioned research served as an inspira- tion for the creation of a new quadratic Lagrange interpolation L1-2 formula. The accuracy of the Caputo-Fabrizio derivative (CFD) was shown to be able to be increased from quadratic convergence to cubic. To the best of our knowledge, no previous L1-2 differentiation formula for the Caputo-Fabrizio derivative (CFD) has been presented before this study. Additionally, this formula’s convergence rate Shankar Pariyar and Jeevan Kafle / BIBECHANA 20 (2023) 126-133 128 has been developed. Even though Caputo Fabrizio’s numerical solution, which makes use of the L1 and L2 numerical formulae, is the main emphasis of this paper, there are other numerical solutions to some specific fractional calculus functions. Gao et al. [13], introduced an innovative fractional numer- ical method, the L1 − 2 formula, for the CFD and used nonlinear interpolation to increase the preci- sion of the numerical calculation at different steps. Our sounds show how the quadratic rate of con- vergence can be transformed into a cubic, even fourth-order, CFD. This study, in our opinion, is the first to support the convergence of the method and to give a new L1−2 differentiation formula for CFD. In addition to studying straightforward frac- tional relaxation and vibration equations, Geren- flo and Mainardi [11] also investigated fractional derivatives. From 1695 to 1697, Leibniz wrote cor- respondence to J. Wallis and J. Bernoulli [17,18] in which he offered a potential method for differenti- ating fractional orders [6]. 2 Preliminaries Some of the priliminaries which are needed for the work are given below : Definition 1 The formula below represents the gamma function’s integral Γ(p) = ∫∞ 0 e−xx(p−1)dx, p > 0. Because only when p > 0 does the afore- mentioned integral converge. The factorial function is also generalized by the gamma function, as for any positive integer p, Γ(p) = ∫∞ 0 e−xx(p−1)dx = (p− 1)! [21]. Definition 2 The beta function, denoted as β(p, q) with p, q > 0, is defined as the integral of xp−1(1− x)q−1 from 0 to 1. It is related to the gamma func- tion through the formula β(p, q) = Γ(p)Γ(q) Γ(p+q) [22]. Definition 3 According to GL, the function M(x) of nth order FD with respect to x is defined by GL if α is a non-negative real number [2]. GL Left sided FD:GLDα a+ [M(x)] = lim n→ 0 1 hα n∑ k=0 (−1)k Γ(α+ 1)M(x− kh) Γ(k + 1)Γ(α− k + 1) , nh = (x− a). . GL Right sided FD:GLDα b− [M(x)] = lim n→ 0 1 hα n∑ k=0 (−1)k Γ(α+ 1)M(x+ kh) Γ(k + 1)Γ(α− k + 1) , nh = (b− x). Definition 4 The R-L fractional Integral, aIxα(m(x)) = 1 Γ(α) ∫ x a (x− z)α−1m(t)dt; x > a. xIaα(m(x)) = 1 Γ(α) ∫ a x (x− z)α−1m(t)dt; a > x. Properties P1: aI α x (K(T (x))) = K ·a Iαx (T (x)). P2: aI α x (P (x)±Q(x))) =a Iαx (P (x))±a I α x (Q(x)). P3: aI α x (N(x)±M(x)) =a Iαx (N(x))±a I α x (M(x)). Definition 5 under Riemann-Liouville, in most cases, the Riemann-Liouville formula defines a fractional derivative [23]. bD q t (v(x)) =  1 Γ(ν−γ) ( d dx )t ∫ q b (q − t)t−γ−1v(t).dt; (t− 1) < β < t dt dxt v(x), β = t Definition 6 The Caputo Fractional derivative is commonly used [13]. aDα x(m(x)) =  1 Γ(n−α) ∫ x a (x− t)n−α−1 ( d dx )n m(t).dt, (n− 1) < α < n dn dxn ·m(x); α = n (1) Definition 7 Mittag - Leffler Function (MLF) of one parameter [10], Eα(z) = ∞∑ n=0 zn Γ(αk + 1) (2) Two-parameters MLF β, R(β) > 0 [11] and Agarwal [24] β, R(β) > 0 Eα,β(z) = ∞∑ n=0 zn Γ(α · k + β) (z, α, β ∈ C) (3) The generation of 3 in terms of series representa- tion was introduced by Prabhakar. Eγ α,β(z) = ∞∑ k=0 (γ)nz k Γ(α · k + β) · n! (4) Where (γ)n is Pochhammer’s symbol. With the aforementioned considerations in mind, we discuss the fractional operators with three-parameter gener- alized Mittag-Leffler kernels in our work. Definition 8 A novel fractional derivative with no singularities in its kernel was suggested by Caputo and Fabrizio in their [17]. The first fractional derivative has an exponential kernel, which is sim- ilar to an exponential function. CF a Dx(f(x)) = M(α) (1−α) ∫ x a e− α(x−t) (1−α) · f ′(t) · dt; M(0) = M(1) = 1 Shankar Pariyar and Jeevan Kafle / BIBECHANA 20 (2023) 126-133 129 Definition 9 Interpolation: The process of figur- ing out the values of the functions at any interme- diate point when the values of the first two points are known is known as interpolation. Definition 10 Linear Interpolation [25]: The lin- ear polynomial that passes through the specified co- ordinates (a0, b0) and (a1, b1). One method to ex- press its formula is as follows: P1(x) = b0 a1−x a1−a0 + b1 x−a0 a1−a0 Definition 11 Quadratic Interpolation [25]: The quadratic polynomial that passes through the spec- ified coordinates (a0, b0), (a1, b1), and (a2, b2) may be written as P2(x) = b0M0(x) + b1M1(x) + b2M2(x), M0(x) = (x− a1)(x− a2) (a0 − a1)(a0 − a2) , M1(x) = (x− a0)(x− a2) (a1 − a0)(a1 − a2) M2(x) = (x− a0)(x− a1) (a2 − a0)(a2 − a1) In the formula above, the polynomial P (x) is known as the Lagrange’s interpolating polynomial, and M0, M1, and M2 are the Lagrange’s interpolating basis functions. 3 Mathematical and numerical meth- ods in Caputo-Fabrizio sense 3.1 L1 Method for the Caputo Frac- tional Derivative Left Caputo fractional derivatives are those frac- tional derivatives of a function that characterize its left-sided behavior, and the exact numerical method used to approximate them is called L1 for 0 < α < 1 C 0 Dα t f(x) ≈ 1 Γ(1− α) ∫ x 0 (x− s)−αf ′(s)ds (5) C 0 Dα t f(xn) ≈ 1 Γ(1− α) ∫ xn 0 (xn − s)−αf ′(s)ds (6) C 0 Dα t f(xn) ≈ 1 Γ(1− α) n−1∑ k=0 ∫ xk+1 xk (xn − s)−αf ′(s)ds Now we replace first-order derivative in 6 by for- ward difference quotient as follows: C 0 Dα t f(xn) ≈ 1 hΓ(1− α) n−1∑ k=0 [f(xk+1 − f(xk)] C 0 Dα t f(xn) ≈ hα Γ(2− α) n−1∑ k=0 [f(xk+1)− f(xk)] [(n− k)(1−α) − (n− k − 1)(1−α)] is known as the L1 approach for Caputo Fractional Derivative. This method is the most popular way to approximate the Caputo fractional derivatives of a function. Using computer tools, we have since con- ducted a more thorough analysis of the numerical method and compare it to the precise solution. 3.2 L2 Method for the Caputo Frac- tional Derivative The left Caputo fractional derivative can be approx- imated via the L2 approach C 0 Dα xf(x) ≈ 1 Γ(2− α) ∫ x 0 (x− s)1−αf ′′(s) · ds, (1 < α < 2) (7)∫ a 0 f(x)dx = ∫ a 0 f(a− x)dx (8) From (7) and (8) as follows, C 0 Dα xf(x) ≈ 1 Γ(2− α) ∫ xn 0 (s)1−αf ′′(xn − s)ds, C 0 Dα xf(xn) ≈ 1 h2(2− α)Γ(2− α) (s)1−α n−1∑ k=0 [f(xn − xk+1)− 2f(xn − xk) + f(xn − xk−1)] [(xk+1) (2−α) − (xk) (2−α)] L2 method for obtaining Caputo fractional deriva- tives is, C 0 Dα xf(xn) ≈ n−1∑ k=0 wkf(xn−k) (9) Such that where wk is the normalizing factor and k has a range of values between −1 and n. It is gen- erally acknowledged that the L2 approach for Ca- puto Fractional Derivative is the preferred method for approximating fractional Caputo derivatives of functions. Recently, a more thorough analysis of this numerical method was done, and the results were compared to the exact ones. 3.3 L1 Method for the Caputo - Fab- rizio Derivative The Caputo Fabrizio Differential operator is defined below for 0 < α < 1 CF 0 Dα xf(x) ≈ M(α) (1− α) ∫ x 0 f ′(s)exp[λ(x− s)]ds, λ = −α (1− α) , (10) CF 0 Dα xf(xn) ≈ M(α) (1− α) ∫ xn 0 f ′(s)exp[λ(xn − s)]ds, (11) Shankar Pariyar and Jeevan Kafle / BIBECHANA 20 (2023) 126-133 130 CF 0 Dα xf(xn) ≈ M(α) (1− α) n∑ k=1 ∫ xk xk−1 f ′(s)exp[λ(xn − s)]ds (12) Now, we replace first-order derivative in (13) by the forward difference quotient as follows: CF 0 Dα xf(xn) ≈ M(α) (1− α) n∑ k=1 ∫ xk xk−1 exp[λ(xn − s)] [ f(xk)− f(xk−1) h ]ds (13) CF 0 Dα xf(xn) = M(α) αh ∑n k=1[f(kh) − f(k − 1)h] [exp[λh(n− k)]− exp[λh(n− k + 1)] which is a derivative of the Caputo Fabrizio equa- tion using the L1 Method. We demonstrate how the three-point backward estimate method, which we derive from the linear Lagrange method, was used to approximate the Caputo Fabrizio deriva- tive. The cubic Lagrange interpolation can be rep- resented with α range of 1 < α < 2. 3.4 L1-2 Method for Caputo Frac- tional Derivative The CFD operator is defined below CF 0 Dα y g(y) = M(α) (1−α) ∫ y 0 g′(u)exp[λ(y−u)]du, λ = −α (1−α) , α ∈ (0, 1). CF 0 Dα y f(yk) = M(α) (1−α) [ ∫ y1 y0 g′(u)exp[λ(yk − u)]du + ∫ y2 y1 g′(u)exp[λ(yk − u)]du + · · · +∫ yk yk−1 g′(u)exp[λ(yk − u)]du], Using the quadratic Lagrange interpolation with the points (yj−2, gj−2), (yj−1, gj−1), (yj , gj) for the backward approximation first order derivative as follows, Π2,jg(y) =gj−2 (y − yj−1)(y − yj) (yj−2 − yj−1)(yj−2 − yj) + gj−1 (y − yj−2)(y − yj) (yj−1 − yj−2)(yj−1 − yj) + gj (y − yj−2)(y − yj−1) (yj − yj−2)(yj − yj−1) Where t ∈ [tj−1, tj ] for 2 ≤ j ≤ k and yj = jh, yj−1 = (j − 1)h. Π2,j(g(y)) = gj−2 2h2 [(y − yj−1)(y − yj)]+ fj−1 h [(x− xj−2)(xj − x) + gj 2h2 [(y − yj−2)(y − yj−1)] Π′ 2j(g(y)) = gj−2 2h2 [(2y − yj−1 − yj)] + gj−1 h [−2y − yj−2 + yj ] + fj 2h2 [2y − yj−2 − yj−1] CF 0 Dα y g(yk) ≈ (g1 − g0) αh (δ1,k − δ0,k) + 1 α k∑ j=2 [ gj−2 2h (1− 2j) + gj−1 h (2j − 2) + gj 2h (3− 2j)(δj,k − δj−1,k)+ gj−2 − 2gj−1 + gj) h2 (yjδj,k)− (1− α) α δj,k − yj−1δj−1,k + 1− α α (δj−1,k) ] 4 Test Examples, Relation between the function f(x) = e−x and its L1 fractional derivative D0.5f(x) Figure 1: L1 Fractional Derivative, D0.5f(x) A continuous blue curve for the function f(x) = e−x and a dashed red curve for D0.5f(x) will be dis- played on the same plot in the two pictures. The function used in the instance, f(x) = e−x, displays an exponential decline. The shape of f(x) is deter- mined by the selected function, f(x) = e−x. The first derivative of the L1 FD, which determines the rate of change, is standardized. Because the fact that the L1 FD shape uses the complete history of the function and is a non-local operator, it is typi- cally smoother and less noisy than the first deriva- tive. The L1 FD is thought to be less that are susceptible to outliers and data noise than other varieties of FD. Example 1 Using the L1 formula and the equation f(x) = sin(x) over [0, 1] at α = 0.5, the numerical results are shown in the table below. Table 1: L1 formula of f(x) = sin(x) at α = 0.5. L1CD* Exact Error Step Sizes 0.8559 0.8461 0.0099 h=0.1 0.8472 0.8461 0.0012 h=0.01 0.8462 0.8461 0.000111 h=0.001 0.8461 0.8461 0.00001074 h=0.0001 0.8461 0.8461 0.0000010611 h=0.00001 Shankar Pariyar and Jeevan Kafle / BIBECHANA 20 (2023) 126-133 131 *L1CD represents L1-Caputo-Derivative. Table 1 displays the numerical outcomes under schemes 5 for the function f(x) = sin(x) over the range [0, 1] us- ing the L1-Caputo-Der formula. There are four columns in the table • L1-Cap-Der: Using the L1-Caputo-Der formula, this column represents a numerical approxima- tion of the integral of the function f(x). • Exact: The value 0.8461 is shown in this column as the integral of the function f(x) over [0, 1]. • Inaccuracy: The absolute inaccuracy between the numerical approximation and the L1-Caputo-Der formula is shown in this column. • Step Sizes: The fourth column labeled "Step Sizes" shows the step size used for each numerical result. From the table, we can see that as the step size de- creases, the numerical result obtained using the L1- Caputo-Der formula gets closer to the exact value of the integral. The error decreases as the step size de- creases, indicating that the numerical approximation is becoming more accurate. At h=0.0001, the error is less than 0.00001, which is a very small error, indicating a highly accurate numerical approximation. Example 2 Using the L2 formula and the equation f(x) = sin(x) over [0, 1] at α = 1.5, the numerical results are shown in the table below. Table 2: L2 formula of f(x) = sin(x) at α = 1.5 L2-CD* Exact Error Step Sizes -0.6219 -0.6697 0.0478 h=0.1 -0.6653 -0.6697 0.0044 h=0.01 -0.6693 -0.6697 0.00042716 h=0.001 -0.6696 -0.6697 0.000042431 h=0.0001 -0.6697 -0.6697 0.0000042307 h=0.00001 L2-CD* represents L2-Caputo Derivative. Table 2 indicates that for the different step sizes to ap- proximate the function f(x) = sin(x) using the L2- Caputo-Derivative, using the scheme 9 over the range [0, 1], α = 1.5 is the value that was utilized in the ap- proximation. The step size has an impact on the nu- merical precision. A smaller step size often results in a more precise estimate, even though it takes longer to calculate. A larger step size results in a less accurate approximation even though it takes less time to com- pute. As a result, while choosing a step size, accuracy and computation effectiveness must be compromised. Example 3 The equation f(x)=cos(4x) over the inter- val [0,2], with alpha=0.1 and different step sizes, can be solved using the L1-formula in the Caputo Fabrizio sense. Table 3: L1 for CFD at f(x) = cos(4x) at α = 0.1 ST*. S.S*. Ex* Ax* Er* 20×101 10−1 -1.0811 -1.0807 3.9720× 10−4 20×102 10−2 -1.0811 -1.0811 3.9720× 10−6 20×103 10−3 -1.0811 -1.0811 3.9720× 10−8 20×104 10−4 -1.0811 -1.0811 3.9720× 10−10 Acronyms: ST* = Steps, S.S* = Steps Sizes, Ex* = Exact, Ax* = Approximate, Er*=Error. If we observe an absolute error reduction of two orders of magnitude while simultaneously reducing the step size by one order of magnitude, this suggests a quadratic rate of convergence at the second order. In other words, not only is our progress significant but it is also hap- pening quite rapidly. Example 4 The equation f(x) = cos(4x) over the in- terval [0, 2], with α = 0.1 and different step sizes, can be solved using the L1 − 2(Backward Approximation) for Caputo-Fractional Derivativeformula in the Caputo Fabrizio sense. Table 4: L1-2 for CFD at f(x) = cos(4x) at α = 0.1 ST*. S.S* Ex* Ax* Er*. 2×101 1× 10−1 -1.0811 -1.0811 2.9514× 10−7 2×102 1× 10−2 -1.0811 -1.0811 2.9090× 10−11 4×101 0.05 -1.0811 -1.0811 1.8377× 10−8 If we observe an absolute error reduction of three or- ders of magnitude while simultaneously reducing the step size by one order of magnitude, this suggests a cubic rate of convergence at the third order. In other words, not only is our progress significant but it is also happening quite rapidly compared to L1-formula in the Caputo-Farbrizio sense. Figure 2: Your figure caption here. When step sizes h1 = 0.1, h2= 0.05, and h = 0.01 are taken in the interval [0, 2], an initial function f(x) = cos(4x) is specified, with a curve to the Shankar Pariyar and Jeevan Kafle / BIBECHANA 20 (2023) 126-133 132 function at each step size. The figure illustrates how, as step size h is reduced, the approximate solutions found using the L1− 2 backwards approximate method approach the exact solution of the equation f(x) = cos(4x).This is so that the function’s derivatives can be approximated with greater accuracy using lower step sizes. 5 Comparative Analysis of a Fractional Derivatives The Grunwald-Letnikov, Riemann-Liouville, Caputo, and L1-2 methods are examples of numerical methods for approximating fractional derivatives. Below is a brief comparison of these two methods: Accuracy: when the fractional derivative’s order is f and the step size is h the convergence rate of the Grunwald-Letnikov method is O(hα). The rate of con- vergence is not as rapid as other methods. • Generally speaking, the Riemann-Liouville tech- nique, which has a convergence rate of O(h2), is more accurate than the Grunwald-Letnikov method. • The Caputo-Fabrizio technique is quicker than the Grunwald-Letnikov and Riemann-Liouville methods, with a convergence rate of O(h2). • The quickest of all the approaches, the L1-2 Ca- puto method converges at a rate of O(h3). Efficiency: • The Grunwald-Letnikov approach only needs to evaluate a discrete convolution, hence it is com- putationally efficient. • The Riemann-Liouville approach can be compu- tationally demanding since it uses numerous in- tegrals. • Utilizing a finite difference formula, the Caputo- Fabrizio technique is computationally effective. • To employ the computationally effective L1-2 Ca- puto approach, the quadratic interpolation for- mula must be assessed. Stability: • The Grunwald-Letnikov method could be numeri- cally unstable depending on the values of the frac- tional derivative order and step size. • The Riemann-Liouville method could be numer- ically unstable for particular values of the frac- tional derivative order and integration limita- tions. • The Caputo-Fabrizio approach is often stable, however it can be sensitive to the selection of step size. • Generally consistent and accurate with the L1-2 caputo approach. Robustness: • The Grunwald-Letnikov method is a usually reli- able technique that can be used to calculate frac- tional derivatives for a variety of functions. • The Riemann-Liouville approach is susceptible to the characteristics of the function being differen- tiated and can be constrained by the selection of integration limits. • The Caputo-Fabrizio method can compute frac- tional derivatives for a variety of functions and is generally reliable. • The L1-2 Caputo method is a usually reliable technique that can be used to calculate fractional derivatives for a variety of functions. The L1-2 Caputo approach is the most precise, effec- tive, stable, and reliable of the four ways, to put it briefly. Although less precise and effective than the L1-2 Caputo method, the Caputo-Fabrizio approach is nevertheless a viable alternative. Compared to Caputo-based approaches, the Grunwald-Letnikov and Riemann-Liouville methods are typically less precise, less effective, and can be vulnerable to numerical in- stabilities. 6 Conclusion Our approach is different from most other domains. A unique numerical approach called as L1, L2, L1 has been devised using the linear and cubic interpolations for the numerical solution of the CFD. It uses the Caputo- Fabrizio sense and L1-2 formula. For backward/forward approximations, The fourth-order convergence rate is found for backward approximations, while the L1 and L1-2 equations provide quadratic and cubic convergence rates. The numerical outcomes were consistent with a well-known functions for different orders and varied step sizes. Every one of these approaches has advan- tages and disadvantages. In contrast to the Riemann- Liouville technique, which can be challenging to com- pute because of singularities, the Grünwald-Letnikov approach is simple to construct but can be slow. 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Anal., pages 1–6, 2013. [24] P. Agarwal and H. Irmak. Some comprehensive in- equalities consisting of mittag-leffler type functions in the complex plane. Mathematical Modelling and Natural Phenomena, 12, 2017. [25] H. S. Malvar, Li-Wei He, and R. Cutler. High- quality linear interpolation for demosaicing of bayer-patterned color images. In 2004 IEEE In- ternational Conference on Acoustics, Speech, and Signal Processing, volume 3, pages iii–485. IEEE, 2004. Introduction Preliminaries Mathematical and numerical methods in Caputo-Fabrizio sense L1 Method for the Caputo Fractional Derivative L2 Method for the Caputo Fractional Derivative L1 Method for the Caputo - Fabrizio Derivative L1-2 Method for Caputo Fractional Derivative Test Examples, Comparative Analysis of a Fractional Derivatives Conclusion