Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 675 https://internationalpubls.com Using W Transform for Solving Volterra Integro-Differential Equations Amal M. Wadi *, Nejmaddin A. Sulaiman ** * ,**Department of Mathematics, College of Education, Salahaddin University-Erbil, Kurdistan Region, Iraq )amal.wadi@su.edu.krd , nejmaddin.sulaiman@su.edu.krd ) Article History: Received: 27-10-2024 Revised: 11-11-2024 Accepted: 19-12-2024 Abstract: There are numerous uses for the Volterra integro-differential equation in the fields of mechanics, geometric probability, population dynamics, theory of rejuvenation, facts on particle size and the damping of string vibration, and transmission of heat issues. Finding the approximate or exact solutions to these equations is of interest to many mathematicians and scientists. Our aim of this paper is to explore and figure out the solution of the Volterra integro-differential equation with a convolution kernel. We now introduce the W transform for determining the solution of linear Volterra integro-differential equation of the second kind and their system. The ability of the W Transform to solve these equations is shown by real- world applications. Based on what we discovered, the W Transform is an effective method for locating precise answers to the linear Volterra integro-differential equation of the second kind and their system. Keywords: Volterra Integro-Differential Equation, W Transform, Convolution, Inverse of W Transform. 1 INTRODUCTION: The study of integral equations possibly considered fundamentally more significant than differential equations since, compared to differential equations, integral equations frequently offer more effective models and less restrictions. Additionally, differentiation in numerical analysis tends to increase error, whereas integration tends to reduce error. The growing use of integral equations in the literature and in many areas of applied mathematics is evidence of their advantages, as stated that certain issues naturally and directly exhibit their mathematical representation, according to integral equations. additional issues, whose direct form is in terms of differential equations, have integral equations more eloquently and compactly substituted for their auxiliary conditions. Integro-differential equations are a common mathematical formulation used to describe physical processes, Several domains, including physics, astronomy, potential theory, fluid dynamics, chemical kinetics, and biological models, use these equations. Multiple techniques have been applied in recent years through certain investigators to solve linear Volterra integro-differential equation of the second kind and their system (LVI-DE of 2nd Kind & LSVI-DE of 2nd kind). We introduce some numerical and analytical methods for solving VI-DE of 2nd Kind & SVI-DE of 2nd kind. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 676 https://internationalpubls.com Taiye Oyedepo et al., presented a computing methodology to deal with VIDEs using shifted Vieta- Lucas polynomials as the foundational basis functions[1].G.A. Aghayeva, et al., Provides a comparison among the mathematical methods used to address the integral Volterra, and ODEs [2]. Asiya Ansari et al., to solve the linear or non-linear VIDE, the Series Solution Method (SSM) is utilized[3]. Ahmad Issa apply the rebuilding of method of Variational Iteration for finding numerical solution of LSVI- DE [4], Mohamed E.A. Alnair and Ahmed A. Khidir are displays a new novel method for resolving LVI-DEs under border restrictions, the technique is founded on combining the Chebyshev spectral techniques [5]. R.A. Olowe et al., By using the multistep collocation method, trigonometry third derivative matched Simpson's method is created and used in order to get close to the answer of VIDEs[6]. O. A. Uwaheren et al., Utilizing Legendre polynomials as basis functions, Akbari-Ganji's Method (AGM) was utilized to solve VIDE[7]. A. Al-Shimmary et al., By using the sixth-order runge- kutta method, they can solve VIDE numerically[8]. Monali Derle and Dinkar Patil , Convolution theory and dual generalization with the purpose of solving IDE, the Rangaig integral transformation had been employed[9]. S. Aggarwal et al., When dealing with LVI-DE of 2nd kind, apply the Sadik transform[10]. S. Al-Ahmad et al., utilize the modified differential transform approach to discover the analytical solution for the LSVI-DE[11]. H. Bozburun and H. A. Peker, apply the Shehu transform to solve the LVI-DE of 2nd kind[12]. N. A. Elbhilil et al., solved the Volterra integral and LVI-DE using the Abaoub-Shkheam transform techniques[13]. H. K. Jassim use the Yang-Laplace transform to determine the analytical solutions for LVI-DE within local fractional operators[14]. Z. Rustam and N. Sulaiman are used (Kamal & SEE) transformation methodology for solving LSVI-DE of 2nd kind[15, 16]. Marjan Uddin and Musafir Uddin, regarding the Laplace transform-based numerical approximation of the LVI-DE[17]. An investigation and use of a recently developed integral transform, or W transform, were conducted by Ping Wang et al. At this point, It has also been established how the W transform relates to other transforms like (Laplace, Sumudu, Natural transform, Elzaki, Mohand, Aboodh, Sawi, Yang, Emad- Falih, Fareeha and Pourreza transform). To show how successful this transformation is, they have solved the differential and integral equations[18]. Kind ndDE of 2-The goal of this work is to use the W transform to quickly and simply solve the LVI kind. ndDE of 2-& LSVI 2 Basic Definitions Definition 1: The LVI-DE of 2nd Kind is provided by [19] : πœ‘(𝑛)(Ο‰) = H(Ο‰) + ∫K(Ο‰, Ο„) Ο†(Ο„)dΟ„ Ο‰ 0 (1) With πœ‘(π‘š)(0) = π‘π‘š , 0 < π‘š < 𝑛 βˆ’ 1 In which the unidentified function πœ‘(Ο„) will be decided, only show up within the integral sign, whereas the derivative of πœ‘(Ο‰) usually take place outside of the sign of integral. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 677 https://internationalpubls.com The Kernels 𝐾(Ο‰, Ο„), and the function H(Ο‰) provided functions with real values, π‘π‘š are constants that define the initial conditions. Definition 2: The general LSVI-DE of 2nd Kind is supplied by [20], πœ‘1 (m)(Ο‰) = H1(Ο‰) + { ∫ K11(Ο‰βˆ’ Ο„) πœ‘1(Ο„)dΟ„+ Ο‰ 0 ∫ K12(Ο‰βˆ’ Ο„)πœ‘2(Ο„)dΟ„ Ο‰ 0 +β‹― .+∫ K1p(Ο‰βˆ’ Ο„)πœ‘p(Ο„)dΟ„ Ο‰ 0 } πœ‘2 (m)(Ο‰) = H2(Ο‰) + { ∫ K21(Ο‰βˆ’ Ο„)πœ‘1(Ο„)dΟ„+ Ο‰ 0 ∫ K22(Ο‰βˆ’ Ο„)πœ‘2(Ο„)dΟ„ Ο‰ 0 +β‹― .+∫ K2p(Ο‰βˆ’ Ο„)πœ‘p(Ο„)dΟ„ Ο‰ 0 } ………………………………………………………………………………. πœ‘p (m)(Ο‰) = Hp(Ο‰) + { ∫ Kn1(Ο‰βˆ’ Ο„)πœ‘1(Ο„)dΟ„+ Ο‰ 0 ∫ Kn2(Ο‰βˆ’ Ο„)πœ‘2(Ο„)dΟ„ Ο‰ 0 +β‹― .+∫ Kpp(Ο‰βˆ’ Ο„)πœ‘p(Ο„)dΟ„ Ο‰ 0 } ] (2) where the unidentified operations πœ‘1(Ο„), πœ‘2(Ο„),… ,πœ‘p(Ο„) that is only show up within the integral symbol, In contrast, the derivatives of πœ‘1(Ο‰), πœ‘2(Ο‰),… , πœ‘p(Ο‰), usually take place outside of the integral sign. The Kernels Kij(Ο‰, Ο„) and Hn(Ο‰) for 𝑖, 𝑗 = 1,2, … , 𝑝 functions with real values. Definition 3: The W transform defined for an exponentially ordered function we examine functions within the set A ,described by A = {πœ‘(Ο‰): βˆƒ M > 0, k > 0, |πœ‘(Ο‰)| < MekΟ‰,Ο‰ ∈ {0,∞}} , the new general integral transform [18], W{πœ‘(Ο‰)} = π‘ π‘š ∫ πœ‘(Ο‰)eβˆ’π‘  𝑛ωdΟ‰ = β„±(s) ∞ 0 (3) where 𝑠 = 𝛼 + 𝑖𝛽. Definition 4: W Transform's inverse for the given function πœ‘(Ο‰) is provided by[18] : π‘Šβˆ’1{β„±(s))} = 1 2Ο€i ∫ 1 π‘ π‘š e𝑠 𝑛ωℱ(s)ds = πœ‘(Ο‰) (4) Ξ±+i∞ Ξ±βˆ’i∞ Table 1: Features of the W Transform S.N Name of Property Form in Mathematics 1. The ability to linearize π‘Š{π‘πœ‘1(Ο‰) + π‘‘πœ‘2(Ο‰)} = π‘π‘Š{πœ‘1(Ο‰)} + π‘‘π‘Š{πœ‘2(Ο‰)} 2. First Derivative π‘Š{πœ‘β€²(Ο‰)} = π‘ π‘›π‘Š(πœ‘(Ο‰)) βˆ’ π‘ π‘š πœ‘(0) 3. Second Derivative π‘Š{πœ‘β€²β€²(Ο‰)} = 𝑠2π‘›π‘Š(πœ‘(Ο‰)) βˆ’ π‘ π‘š+𝑛 πœ‘(0) βˆ’ π‘ π‘š πœ‘β€²(0) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 678 https://internationalpubls.com 4. nth Derivative π‘Š{πœ‘(k)(Ο‰)} = π‘ π‘˜π‘›π‘Š(πœ‘(Ο‰)) βˆ’ π‘ π‘šβˆ‘π‘ π‘›(π‘˜βˆ’π‘—βˆ’1) π‘˜βˆ’1 𝑗=0 πœ‘(𝑗)(0), π‘˜ β‰₯ 1 5. Convolution π‘Š{πœ‘1(Ο‰) βˆ— πœ‘2(Ο‰) } = 1 π‘ π‘š π‘Š{πœ‘1(Ο‰)} βˆ— π‘Š{πœ‘2(Ο‰) } Table 2: W Transform of useful Functions S.N Ο†(Ο‰) π‘Š{Ο†(Ο‰)} = β„±(s) 1 c 𝑐 π‘ π‘š 𝑠𝑛 2. Ο‰ π‘ π‘š 𝑠2𝑛 3. aΟ‰π‘˜ π‘Ž π‘˜! π‘ π‘š 𝑠𝑛(π‘˜+1) , a constant .4 π‘’π‘ŽΟ‰ π‘ π‘š 𝑠𝑛 βˆ’ π‘Ž 5. sin (π‘ŽΟ‰) π‘Ž π‘ π‘š 𝑠2𝑛 + π‘Ž2 6. cos (π‘ŽΟ‰) π‘ π‘š+𝑛 𝑠2𝑛 + π‘Ž2 7. sinh (π‘ŽΟ‰) π‘Ž π‘ π‘š 𝑠2𝑛 βˆ’ π‘Ž2 8. cosh (π‘ŽΟ‰) π‘ π‘š+𝑛 𝑠2𝑛 βˆ’ π‘Ž2 3 Methodology Within this part, We introduce the W transform as a means of solving LVI-DE of 2nd Kind and LSVI-DE of 2nd kind. In this piece of writing, Assumed to be the kernel K(Ο‰, Ο„) can be expressed by difference, which is (Ο‰βˆ’ Ο„). I.Use W Transform to Resolve LVI-DE of 2nd Kind: πœ‘(𝑛)(Ο‰) = H(Ο‰) + ∫ K(Ο‰βˆ’ Ο„) Ο†(Ο„)dΟ„ Ο‰ 0 (5) With Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 679 https://internationalpubls.com πœ‘(π‘š)(0) = π‘π‘š , 0 < π‘š < 𝑛 βˆ’ 1 (6) Taking W transform's on (5) and making use of convolution theorem, we get π‘Š{πœ‘(𝑛)(Ο‰)} = π‘Š{H(Ο‰)} + 1 π‘ π‘š π‘Š{K(Ο‰)}π‘Š{Ο†(Ο‰)} (7) Making use of the property β€œW transforms of derivatives” on (7), We obtain π‘ π‘˜π‘›π‘Š{πœ‘(Ο‰)} βˆ’ π‘ π‘š βˆ‘ 𝑠𝑛(π‘˜βˆ’π‘—βˆ’1)π‘˜βˆ’1 𝑗=0 πœ‘(𝑗)(0) = π‘Š{H(Ο‰)} + 1 π‘ π‘š π‘Š{K(Ο‰)}π‘Š{Ο†(Ο‰)} (8) substituting equation (6) in equation (8), after simplification of equation (8), and we have the values of π‘Š{H(Ο‰)},π‘Š{K(Ο‰)} . following the inverse W transform according to these principles, We receive the necessary value of πœ‘(Ο‰). II.Use W Transform to Resolve LSVI-DE of 2 nd kind: πœ‘1 (k)(Ο‰) = H1(Ο‰) + { ∫ K11(Ο‰βˆ’ Ο„) πœ‘1(Ο„)dΟ„+ Ο‰ 0 ∫ K12(Ο‰βˆ’ Ο„)πœ‘2(Ο„)dΟ„ Ο‰ 0 +β‹― .+∫ K1p(Ο‰βˆ’ Ο„)πœ‘p(Ο„)dΟ„ Ο‰ 0 } πœ‘2 (k)(Ο‰) = H2(Ο‰) + { ∫ K21(Ο‰βˆ’ Ο„)πœ‘1(Ο„)dΟ„+ Ο‰ 0 ∫ K22(Ο‰βˆ’ Ο„)πœ‘2(Ο„)dΟ„ Ο‰ 0 +β‹― .+∫ K2p(Ο‰βˆ’ Ο„)πœ‘p(Ο„)dΟ„ Ο‰ 0 } ………………………………………………………………………………. πœ‘p (k)(Ο‰) = Hp(Ο‰) + { ∫ Kp1(Ο‰βˆ’ Ο„)πœ‘1(Ο„)dΟ„+ Ο‰ 0 ∫ Kp2(Ο‰βˆ’ Ο„)πœ‘2(Ο„)dΟ„ Ο‰ 0 +β‹― .+∫ Kpp(Ο‰βˆ’ Ο„)πœ‘p(Ο„)dΟ„ Ο‰ 0 } ] (9) With { πœ‘ 1 (j)(0) = a1j, j = 0,1,2, … , kβˆ’ 1; πœ‘ 2 (j)(0) = a2j, j = 0,1,2, … , kβˆ’ 1; …………………………………………… πœ‘p (j)(0) = apj, j = 0,1,2, … , kβˆ’ 1; } (10) Taking W transformation proprietor on system (9) ,then applying the convolution theorem, we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 680 https://internationalpubls.com π‘Š{πœ‘1 (k)(Ο‰)} = π‘Š{H1(Ο‰)} + [ 1 π‘ π‘š π‘Š{𝐾11(Ο‰)}π‘Š{ πœ‘1(Ο‰)} + 1 π‘ π‘š π‘Š{𝐾12(Ο‰)}π‘Š{ πœ‘2(Ο‰)} +β‹―+ 1 π‘ π‘š π‘Š{𝐾1𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} ] π‘Š{πœ‘2 (k)(Ο‰)} = π‘Š{H2(Ο‰)} + [ 1 π‘ π‘š π‘Š{𝐾21(Ο‰)}π‘Š{ πœ‘1(Ο‰)} + 1 π‘ π‘š π‘Š{𝐾22(Ο‰)}π‘Š{ πœ‘2(Ο‰)} +β‹―+ 1 π‘ π‘š π‘Š{𝐾2𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} ] ………………………………… .………………………………………………… π‘Š{πœ‘p (k)(Ο‰)} = π‘Š{Hp(Ο‰)} + [ 1 π‘ π‘š π‘Š{𝐾𝑝1(Ο‰)}π‘Š{ πœ‘1(Ο‰)} + 1 π‘ π‘š π‘Š{𝐾𝑝2(Ο‰)}π‘Š{ πœ‘2(Ο‰)} +β‹―+ 1 π‘ π‘š π‘Š{𝐾𝑝𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} ] ] (11) Making use of the property β€œW transforms of derivatives” on system (11), we get { π‘ π‘˜π‘›π‘Š( πœ‘1(Ο‰)) βˆ’π‘ π‘šβˆ‘π‘ π‘›(π‘˜βˆ’π‘—βˆ’1) π‘˜βˆ’1 𝑗=0 πœ‘1 (𝑗)(0) } = π‘Š{H1(Ο‰)} + [ 1 π‘ π‘š π‘Š{𝐾11(Ο‰)}π‘Š{ πœ‘1(Ο‰)} + 1 π‘ π‘š π‘Š{𝐾12(Ο‰)}π‘Š{ πœ‘2(Ο‰)} +β‹―+ 1 π‘ π‘š π‘Š{𝐾1𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} ] { π‘ π‘˜π‘›π‘Š( πœ‘2(Ο‰)) βˆ’π‘ π‘šβˆ‘π‘ π‘›(π‘˜βˆ’π‘—βˆ’1) π‘˜βˆ’1 𝑗=0 πœ‘2 (𝑗)(0) } = π‘Š{H2(Ο‰)} + [ 1 π‘ π‘š π‘Š{𝐾21(Ο‰)}π‘Š{ πœ‘1(Ο‰)} + 1 π‘ π‘š π‘Š{𝐾22(Ο‰)}π‘Š{ πœ‘2(Ο‰)} +β‹―+ 1 π‘ π‘š π‘Š{𝐾2𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} ] ……… . . ……………………………………………………………………………………… { π‘ π‘˜π‘›π‘Š( πœ‘p(Ο‰)) βˆ’π‘ π‘šβˆ‘π‘ π‘›(π‘˜βˆ’π‘—βˆ’1) π‘˜βˆ’1 𝑗=0 πœ‘p (𝑗)(0) } = π‘Š{Hp(Ο‰)} + [ 1 π‘ π‘š π‘Š{𝐾𝑝1(Ο‰)}π‘Š{ πœ‘1(Ο‰)} + 1 π‘ π‘š π‘Š{𝐾𝑝2(Ο‰)}π‘Š{ πœ‘2(Ο‰)} +β‹―+ 1 π‘ π‘š π‘Š{𝐾𝑝𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} ] ] (12) substituting initial condition (10) in system (12), we obtain Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 681 https://internationalpubls.com { 𝑠 π‘˜π‘›π‘Š( πœ‘1(Ο‰)) βˆ’π‘ π‘š+𝑛(π‘˜βˆ’1)π‘Ž10 βˆ’π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž11 βˆ’β‹―βˆ’ π‘ π‘šπ‘Ž1π‘˜βˆ’1} = π‘Š{H1(Ο‰)} + [ 1 π‘ π‘š π‘Š{𝐾11(Ο‰)}π‘Š{ πœ‘1(Ο‰)} + 1 π‘ π‘š π‘Š{𝐾12(Ο‰)}π‘Š{ πœ‘2(Ο‰)} +β‹―+ 1 π‘ π‘š π‘Š{𝐾1𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} ] { 𝑠 π‘˜π‘›π‘Š( πœ‘2(Ο‰)) βˆ’π‘ π‘š+𝑛(π‘˜βˆ’1)π‘Ž20 βˆ’π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž21 βˆ’β‹―βˆ’ π‘ π‘šπ‘Ž2π‘˜βˆ’1} = π‘Š{H2(Ο‰)} + [ 1 π‘ π‘š π‘Š{𝐾21(Ο‰)}π‘Š{ πœ‘1(Ο‰)} + 1 π‘ π‘š π‘Š{𝐾22(Ο‰)}π‘Š{ πœ‘2(Ο‰)} +β‹―+ 1 π‘ π‘š π‘Š{𝐾2𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} ] ……… . . …………………………………………………………………………………… { 𝑠 π‘˜π‘›π‘Š( πœ‘π‘(Ο‰)) βˆ’π‘ π‘š+𝑛(π‘˜βˆ’1)π‘Žπ‘0 βˆ’π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Žπ‘1 βˆ’β‹―βˆ’ π‘ π‘šπ‘Žπ‘π‘˜βˆ’1} = π‘Š{Hp(Ο‰)} + [ 1 π‘ π‘š π‘Š{𝐾𝑝1(Ο‰)}π‘Š{ πœ‘1(Ο‰)} + 1 π‘ π‘š π‘Š{𝐾𝑝2(Ο‰)}π‘Š{ πœ‘2(Ο‰)} +β‹―+ 1 π‘ π‘š π‘Š{𝐾𝑝𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} ] ] (13) After simplification system (13), we get { [π‘ π‘˜π‘› βˆ’ 1 π‘ π‘š π‘Š{𝐾11(Ο‰)}] π‘Š{ πœ‘1(Ο‰)} βˆ’ 1 π‘ π‘š π‘Š{𝐾12(Ο‰)}π‘Š{ πœ‘2(Ο‰)} βˆ’β‹―βˆ’ 1 π‘ π‘š π‘Š{𝐾1𝑝(Ο‰)}π‘Š{ πœ‘π‘(Ο‰)} } = [ π‘Š{H1(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Ž10 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž11 +β‹―+π‘ π‘šπ‘Ž1π‘˜βˆ’1 ] { + βˆ’ 1 π‘ π‘š π‘Š{𝐾21(Ο‰)}π‘Š{ πœ‘1(Ο‰)} [ π‘ π‘˜π‘› βˆ’ 1 π‘ π‘š π‘Š{𝐾22(Ο‰)}] π‘Š{ πœ‘2(Ο‰)} βˆ’β‹―βˆ’ 1 π‘ π‘š π‘Š{𝐾2𝑝(Ο‰)}π‘Š{ πœ‘p(Ο‰)} } = [ π‘Š{H2(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Ž20 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž21 +β‹―+ π‘ π‘šπ‘Ž2π‘˜βˆ’1 ] …… . . ……………………………………………………………………………… { βˆ’1 π‘ π‘š π‘Š{𝐾𝑝1(Ο‰)}π‘Š{ πœ‘1(Ο‰)} βˆ’ 1 π‘ π‘š π‘Š{𝐾𝑝2(Ο‰)}π‘Š{ πœ‘2(Ο‰)} βˆ’β‹―+ [π‘ π‘˜π‘› βˆ’ 1 π‘ π‘š π‘Š{𝐾𝑝𝑝(Ο‰)}] π‘Š{ πœ‘p(Ο‰)}} = [ π‘Š{Hp(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Žπ‘0 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Žπ‘1 +β‹―+π‘ π‘šπ‘Žπ‘π‘˜βˆ’1 ] ] (14) The solution of system (14) is given as Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 682 https://internationalpubls.com π‘Š{ πœ‘1(Ο‰)} = | | | | { π‘Š{H1(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Ž10 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž11 +β‹―+π‘ π‘šπ‘Ž1π‘˜βˆ’1 } βˆ’ 1 π‘ π‘šπ‘Š {𝐾12(Ο‰)} ……… βˆ’ 1 π‘ π‘šπ‘Š {𝐾1𝑝(Ο‰)} { π‘Š{H2(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Ž20 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž21 +β‹―+ π‘ π‘šπ‘Ž2π‘˜βˆ’1 } ( π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾22(Ο‰)}) ……… βˆ’ 1 π‘ π‘šπ‘Š {𝐾2𝑝(Ο‰)} …………………………………………………………… { π‘Š{Hp(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Žπ‘0 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Žπ‘1 +β‹―+π‘ π‘šπ‘Žπ‘π‘˜βˆ’1 } βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝2(Ο‰)} ……… (𝑠 π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝𝑝(Ο‰)})| | | | | | | [π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾11(Ο‰)}] βˆ’ 1 π‘ π‘šπ‘Š {𝐾12(Ο‰)} ……… βˆ’ 1 π‘ π‘šπ‘Š {𝐾1𝑝(Ο‰)} βˆ’ 1 π‘ π‘šπ‘Š {𝐾21(Ο‰)} [ π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾22(Ο‰)}] ……… βˆ’ 1 π‘ π‘šπ‘Š {𝐾2𝑝(Ο‰)} ………………………………………… βˆ’1 π‘ π‘šπ‘Š {𝐾𝑝1(Ο‰)} βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝2(Ο‰)} ……… [π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝𝑝(Ο‰)}] | | | π‘Š{ πœ‘2(Ο‰)} = | | | | [π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾11(Ο‰)}] { π‘Š{H1(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Ž10 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž11 +β‹―+π‘ π‘šπ‘Ž1π‘˜βˆ’1 } ……… βˆ’ 1 π‘ π‘šπ‘Š {𝐾1𝑝(Ο‰)} βˆ’ 1 π‘ π‘šπ‘Š {𝐾21(Ο‰)} { π‘Š{H2(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Ž20 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž21 +β‹―+ π‘ π‘šπ‘Ž2π‘˜βˆ’1 } … … … βˆ’ 1 π‘ π‘šπ‘Š {𝐾2𝑝(Ο‰)} …………………………………………………………… βˆ’1 π‘ π‘šπ‘Š {𝐾𝑝1(Ο‰)} { π‘Š{Hp(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Žπ‘0 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Žπ‘1 +β‹―+π‘ π‘šπ‘Žπ‘π‘˜βˆ’1 } ……… (π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝𝑝(Ο‰)})| | | | | | | [π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾11(Ο‰)}] βˆ’ 1 π‘ π‘šπ‘Š {𝐾12(Ο‰)} ……… βˆ’ 1 π‘ π‘šπ‘Š {𝐾1𝑝(Ο‰)} βˆ’ 1 π‘ π‘šπ‘Š {𝐾21(Ο‰)} [ π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾22(Ο‰)}] ……… βˆ’ 1 π‘ π‘šπ‘Š {𝐾2𝑝(Ο‰)} ……………………………………… βˆ’1 π‘ π‘šπ‘Š {𝐾𝑝1(Ο‰)} βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝2(Ο‰)} ……… [π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝𝑝(Ο‰)}] | | | Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 683 https://internationalpubls.com ………………………………………………………………………………………… π‘Š{ πœ‘p(Ο‰)} = | | | | [𝑠 π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾11(Ο‰)}] βˆ’ 1 π‘ π‘šπ‘Š {𝐾12(Ο‰)} ……… { π‘Š{H1(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Ž10 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž11 +β‹―+π‘ π‘šπ‘Ž1π‘˜βˆ’1 } βˆ’ 1 π‘ π‘šπ‘Š {𝐾21(Ο‰)} [ 𝑠 π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾22(Ο‰)}] ……… { π‘Š{H2(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Ž20 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Ž21 +β‹―+ π‘ π‘šπ‘Ž2π‘˜βˆ’1 } …………………………………………………………… βˆ’1 π‘ π‘šπ‘Š {𝐾𝑝1(Ο‰)} βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝2(Ο‰)} ……… { π‘Š{Hp(Ο‰)} + 𝑠 π‘š+𝑛(π‘˜βˆ’1)π‘Žπ‘0 +π‘ π‘š+𝑛(π‘˜βˆ’2)π‘Žπ‘1 +β‹―+π‘ π‘šπ‘Žπ‘π‘˜βˆ’1 } | | | | | | | [π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾11(Ο‰)}] βˆ’ 1 π‘ π‘šπ‘Š {𝐾12(Ο‰)} ……… βˆ’ 1 π‘ π‘šπ‘Š {𝐾1𝑝(Ο‰)} βˆ’ 1 π‘ π‘šπ‘Š {𝐾21(Ο‰)} [ π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾22(Ο‰)}] ……… βˆ’ 1 π‘ π‘šπ‘Š {𝐾2𝑝(Ο‰)} ………………………………………… βˆ’1 π‘ π‘šπ‘Š {𝐾𝑝1(Ο‰)} βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝2(Ο‰)} ……… [π‘ π‘˜π‘› βˆ’ 1 π‘ π‘šπ‘Š {𝐾𝑝𝑝(Ο‰)}] | | | When the aforementioned equations are simplified, we obtain the values of π‘Š{ πœ‘1(Ο‰)},π‘Š{ πœ‘2(Ο‰)},… ,π‘Š{ πœ‘p(Ο‰)}. Following the inverse W transform according to these principles, We get the necessary values of πœ‘1(Ο‰), πœ‘2(Ο‰),… , πœ‘p(Ο‰). 4 NUMERICAL PROBLEMS: This section presents several applications that illustrate the efficiency of the W transform in resolving LVI-DE of 2nd Kind and LSVI-DE of 2nd kind. Problem 1: Consider the LVI-DE of 2nd Kind. πœ‘β€²(Ο‰) = 1βˆ’βˆ« Ο†(Ο„)dΟ„ Ο‰ 0 (15) with Ο†(0) = 0 (16) Operating w transform on equation (15) π‘Š{πœ‘β€²(Ο‰)} = π‘Š{1} βˆ’ π‘Š{∫ Ο†(Ο„)dΟ„ } Ο‰ 0 (17) as well as employing the convolution theorem, We've got π‘Š{πœ‘β€²(Ο‰)} = π‘Š{1} βˆ’ 1 π‘ π‘š π‘Š{1}π‘Š{Ο†(Ο‰)} (18) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 684 https://internationalpubls.com Utilizing the asset β€œW transforms of derivatives” in equation (18), We own π‘ π‘›π‘Š{Ο†(Ο‰)} βˆ’ π‘ π‘šΟ†(0) = π‘ π‘š 𝑠𝑛 βˆ’ 1 π‘ π‘š ( π‘ π‘š 𝑠𝑛 )π‘Š{Ο†(Ο‰)} (19) substituting initial condition (16) in (19), after simplification of equation (19) and operating inverse W transforms, we get the required solution of equation Ο†(Ο‰) = sin(Ο‰) (20) Problem 2: Consider the LVI-DE of 2nd Kind. Ο†β€²β€²(Ο‰) = 1+ ∫ (Ο‰βˆ’ Ο„) Ο†(Ο„)dΟ„ Ο‰ 0 (21) with Ο†(0) = 1, Ο†β€²(0) = 0 (22) Utilizing the W transform to each side of (21), We've got π‘Š{Ο†β€²β€²(Ο‰)} = π‘Š{1} +π‘Š{∫ (Ο‰βˆ’ Ο„) Ο†(Ο„)dΟ„ Ο‰ 0 } (23) as well as employing the convolution theorem on equation (23), We obtain π‘Š{Ο†β€²β€²(Ο‰)} = π‘Š{1} + 1 π‘ π‘š π‘Š{Ο‰} π‘Š{Ο†(Ο‰)} (24) Making use of the asset β€œw transforms of derivatives” on equation (24), We own 𝑠2π‘›π‘Š{Ο†(Ο‰)} βˆ’ π‘ π‘š+𝑛φ(0) βˆ’ π‘ π‘šΟ†β€²(0) = π‘ π‘š 𝑠𝑛 + 1 π‘ π‘š ( π‘ π‘š 𝑠2𝑛 )π‘Š{Ο†(Ο‰)} (25) substituting initial condition (22) in (25) ,after simplification of equation (25) and operating inverse W transforms, we get the required solution of equation Ο†(Ο‰) = cosh(Ο‰) (26) Problem 3: Consider the LSVI-DE of 2nd kind. { πœ‘1 β€²(Ο‰) = 2Ο‰2 +∫ [(Ο‰βˆ’ Ο„)πœ‘1(Ο„) + (Ο‰βˆ’ Ο„)πœ‘2(Ο„)]dΟ„ Ο‰ 0 πœ‘2 β€²(Ο‰) = βˆ’3Ο‰2 βˆ’ 1 10 Ο‰5 +∫ [(Ο‰βˆ’ Ο„)πœ‘1(Ο„) βˆ’ (Ο‰βˆ’ Ο„)πœ‘2(Ο„)]dΟ„ Ο‰ 0 } (27) with πœ‘1(0) = 1 , πœ‘2(0) = 1 (28) Implementing the W transform to both parties of (27), We've got { π‘Š{πœ‘1 β€²(Ο‰)} = W{2Ο‰2} +W{∫ [(Ο‰βˆ’ Ο„)πœ‘1(Ο„) + (Ο‰βˆ’ Ο„)πœ‘2(Ο„)]dΟ„ } Ο‰ 0 π‘Š{πœ‘2 β€²(Ο‰)} = W{βˆ’3Ο‰2} βˆ’ W { 1 10 Ο‰5} + π‘Š{∫ [(Ο‰βˆ’ Ο„)πœ‘1(Ο„) βˆ’ (Ο‰βˆ’ Ο„)πœ‘2(Ο„)]dΟ„} Ο‰ 0 } (29) Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 685 https://internationalpubls.com and using the convolution theorem of the W transform on equation (29), we get { π‘Š{πœ‘1 β€²(Ο‰)} = W{2Ο‰2} + 1 π‘ π‘š [ W{Ο‰}W{πœ‘1(Ο‰)} +W{Ο‰}W{πœ‘2(Ο‰)}] π‘Š{πœ‘2 β€²(Ο‰)} = W{βˆ’3Ο‰2} βˆ’W { 1 10 Ο‰5} + 1 π‘ π‘š [ W{Ο‰}W{πœ‘1(Ο‰)} βˆ’ W{Ο‰}W{πœ‘2(Ο‰)}] } (30) Making use of the asset β€œW transforms of derivatives” on equation (30), We've got { π‘ π‘›π‘Š{πœ‘1(Ο‰)} βˆ’ 𝑠 π‘šΟ† 1 (0) = 2( 2! π‘ π‘š 𝑠3𝑛 ) + 1 π‘ π‘š [ π‘ π‘š 𝑠2𝑛 W{πœ‘1(Ο‰)} + π‘ π‘š 𝑠2𝑛 W{πœ‘2(Ο‰)}] π‘ π‘›π‘Š{πœ‘2(Ο‰)} βˆ’ 𝑠 π‘šΟ† 2 (0) = βˆ’3( 2! π‘ π‘š 𝑠3𝑛 ) βˆ’ 1 10 ( 5! π‘ π‘š 𝑠6𝑛 ) + 1 π‘ π‘š [ π‘ π‘š 𝑠2𝑛 W{πœ‘1(Ο‰)} βˆ’ π‘ π‘š 𝑠2𝑛 W{πœ‘2(Ο‰)}] } (31) substituting initial condition (28) in system (31)and after simplification we get { ( 𝑠3𝑛 βˆ’ 1 𝑠2𝑛 ) π‘Š{πœ‘1(Ο‰)} βˆ’ ( 1 𝑠2𝑛 ) π‘Š{πœ‘2(Ο‰)} = 4π‘ π‘š + π‘ π‘š+3𝑛 𝑠3𝑛 ( βˆ’1 𝑠2𝑛 ) π‘Š{πœ‘1(Ο‰)} + ( 𝑠3𝑛 + 1 𝑠2𝑛 ) π‘Š{πœ‘2(Ο‰)} = βˆ’6π‘ π‘š+3𝑛 βˆ’ 12π‘ π‘š + π‘ π‘š+6𝑛 𝑠6𝑛 } (32) The solution of system (32) is given as π‘Š{πœ‘1(Ο‰)} = | 4π‘ π‘š + π‘ π‘š+3𝑛 𝑠3𝑛 βˆ’ 1 𝑠2𝑛 βˆ’6π‘ π‘š+3𝑛 βˆ’ 12π‘ π‘š + π‘ π‘š+6𝑛 𝑠6𝑛 𝑠3𝑛 + 1 𝑠2𝑛 | | 𝑠3𝑛 βˆ’ 1 𝑠2𝑛 βˆ’ 1 𝑠2𝑛 βˆ’1 𝑠2𝑛 𝑠3𝑛 + 1 𝑠2𝑛 | π‘Š{πœ‘2(Ο‰)} = | 𝑠3𝑛 βˆ’ 1 𝑠2𝑛 4π‘ π‘š + π‘ π‘š+3𝑛 𝑠3𝑛 βˆ’1 𝑠2𝑛 βˆ’6π‘ π‘š+3𝑛 βˆ’ 12π‘ π‘š + π‘ π‘š+6𝑛 𝑠6𝑛 | | 𝑠3𝑛 βˆ’ 1 𝑠2𝑛 βˆ’ 1 𝑠2𝑛 βˆ’1 𝑠2𝑛 𝑠3𝑛 + 1 𝑠2𝑛 | When the aforementioned equations are simplified, we obtain the values of π‘Š{ πœ‘1(Ο‰)} = π‘ π‘š 𝑠𝑛 + 6 π‘ π‘š 𝑠4𝑛 π‘Š{ πœ‘2(Ο‰)} = π‘ π‘š 𝑠𝑛 βˆ’ 6 π‘ π‘š 𝑠4𝑛 Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 686 https://internationalpubls.com operating inverse W transforms, we get the required solution of equations πœ‘1(Ο‰) = 1+ Ο‰3 πœ‘2(Ο‰) = 1βˆ’ Ο‰3 5 Conclusion: In this research, we have effectively addressed the W transform for resolving LVI-DE of 2nd Kind and LSVI-DE of 2nd kind, and we have extensively detailed the process by considering three numerical problems. The answers to these issues show how beneficial and efficient the W transform in whitening the resolving LVI-DE of 2nd Kind and LSVI-DE of 2nd kind. The provided applications demonstrate that a precise solution was found in a very short amount of time and with very little processing power. REFERENCES [1] Oyedepo, T., A.M. Ayinde, and E.N. Didigwu, VIETA-LUCAS POLYNOMIAL COMPUTATIONAL TECNIQUE FOR VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS. Electronic Journal of Mathematical Analysis and Applications, 2024. 12(1): p. 1-8. [2] Aghayeva, G., V. Ibrahimov, and D. Juraev, ON SOME COMPARISION OF THE NUMERICAL METHODS APPLIED TO SOLVE ODES, VOLTERRA INTEGRAL AND INTEGRO DIFFERENTIAL EQUATIONS. Karshi Multidisciplinary International Scientific Journal, 2024. 1(1). [3] ANSARI, A., N. AHMAD, and A.H. ALI, NUMERICAL STUDY OF THE SERIES SOLUTION METHOD TO ANALYSIS OF VOLTERRA INTEGRO-DIFFERENTIAL EQUATIONS. Journal of applied mathematics & informatics, 2024. 42(4): p. 899-913. [4] Issa, A., Numerical Solution of System of Linear Volterra Integro-Differential Equations by Reconstruction of Variational Iteration Method. Palestine Journal of Mathematics, 2023. 12(4). [5] Alnair, M.E. and A.A. Khidir. Approximation Technique for Solving Linear Volterra Integro‐Differential Equations with Boundary Conditions. in Abstract and Applied Analysis. 2022. Wiley Online Library. [6] Olowe, R., et al., Trigonometrically-Fitted Simpson’s Method for Solving Volterra Integro-Differential Equations. International Journal of Mathematical Sciences and Optimization: Theory and Applications, 2022. 8(2): p. 68-78. [7] Uwaheren, O., et al., Numerical Solution of Volterra integro-differential Equations by Akbari-Ganji’s Method. BAREKENG: Jurnal Ilmu Matematika dan Terapan, 2022. 16(3): p. 1123-1130. [8] Al-Shimmary, A., A. Hussain, and S. Radhi. Numerical solution of volterra integro–differential equation using 6th order runge-kutta method. in Journal of Physics: Conference Series. 2021. IOP Publishing. [9] Derle, M. and D. Patil, Applications of The Double General Rangaig Integral Transform in Integro-Differential Equations. Indian Journal of Science and Technology, 2024. 17(31): p. 3258-3271. [10] Aggarwal, S., A. Gupta, and S. Sharma, Application of Sadik transform for handling linear Volterra integro- differential equations of second kind. Universal Review, 2019. 10(7): p. 177-187. [11] Al-Ahmad, S., et al., Analytical solution of systems of Volterra integro-differential equations using modified differential transform method. J. Math. Comput. Sci, 2022. 26: p. 1-9. [12] Bozburun, H. and H.A. Peker, Solution of linear Volterra integro-differential equations of second kind using Shehu transform. Advanced Studies: Euro-Tbilisi Mathematical Journal, 2021: p. 177-186. [13] Elbhilil, N.A., M.I. Bnis, and A.A. Altirban, Abaoub-Shkheam Transform Techniques to Solve Volterra Integral and Volterra Integro-Differential Equations. African Journal of Advanced Pure and Applied Sciences (AJAPAS), 2023: p. 254-260. [14] Jassim, H.K., The analytical solutions for volterra integro-differential equations within local fractional operators by yang-laplace transform. Sahand Communications in Mathematical Analysis, 2017. 6(1): p. 69-76. [15] Rustam, Z. and N. Sulaiman, Kamal transform technique for solving system of linear Volterra integro-differential equations of the second kind. International Journal of Nonlinear Analysis and Applications, 2023. 14(1): p. 185- 192. [16] Rustam, Z.R. and N.A. Sulaiman, SEE Transform Technique for Solving System of Linear Volterra Integro- Differential Equations of the Second Kind. Polytechnic Journal, 2023. 12(2): p. 2. Communications on Applied Nonlinear Analysis ISSN: 1074-133X Vol 32 No. 7s (2025) 687 https://internationalpubls.com [17] Uddin, M. and M. Uddin, On the numerical approximation of Volterra integro-differential equation using Laplace transform. Computational Methods for Differential Equations, 2020. 8(2): p. 305-313. [18] Wang, P., X.-Y. Peng, and F. Wang, The analysis and application of a new integral transform W transform. Thermal Science, 2023. 27(5 Part A): p. 3823-3827. [19] Wazwaz, A.-M., Linear and nonlinear integral equations. Vol. 639. 2011: Springer. 20. Aggarwal, S. and S. Kumar, Solution of system of linear Volterra integro-differential equations of second kind via Laplace-Carson transform. J. Emerg. Technol. Innov. Res, 2021. 8.