Solutions of Integral Equations by using Successive Approximation Method

Abstract Background/Objectives: Integral transforms and iterative techniques play a central role in the theory and solution of integral equations. In this work, generalized special functions, namely Prefunctions and Extended Prefunctions are used to obtain exact and approximate solutions of Volterra integral equations of the second kind. These functions are parameterized by 𝛼 and extend the analytic structure of exponential, hyperbolic and trigonometric functions. These functions are effective tools in solving differential and integral equations. Methods: In this study successive approximations method is used to derive closed-form solutions for two classes of Volterra integral equations with different kernels. For the kernel 𝐾(𝑥, 𝑡) = 𝑥 − 𝑡 with 𝑓 (𝑥, 𝛼) = 𝑝𝑠𝑖𝑛(𝑥, 𝛼) the iterative sequence yields a generalized solution expressed as a series of shifted Prefunctions, reducing to 𝑦(𝑥, 0) = 1 2 (𝑝𝑠𝑖𝑛ℎ(𝑥) + 𝑠𝑖𝑛(𝑥)) for 𝛼 =0 confirming the consistency of the method with known classical results. For the convolution kernel 𝐾(𝑥, 𝑡) = 𝑒−(𝑥−𝑠) with 𝑓(𝑥, 𝛼) = 𝑝𝑠𝑖𝑛ℎ(𝑥, 𝛼) the Picard iteration process leads to the compact representation 𝑦(𝑥, 𝛼) = 𝑝𝑠𝑖𝑛ℎ(𝑥, 𝛼) + 𝑠𝑖𝑛(𝑥, 𝛼 + 1). Findings: Numerical illustrations and graphical results confirm uniform convergence of the successive approximations to the exact solution for various values of 𝛼. For a large value of 𝛼 convergence rate is higher. Novelty: Prefunction -based methods provide generalized frameworks for solving Volterra integral equations and open further directions for applications in mathematical physics and engineering models. 2020 Mathematics Subject Classification: 45Dxx, 34A12. Keywords: Integral Equations; Prefunctions; Successive Approximation Method; Convergence; Picard Iterates