Advanced Modified Iterative Techniques for Integral Transform Solutions of Differential Equations using as Laplace, Sumudu, and Elzaki

Document Type : Original Article

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Department of information systems, Higher institute for computers Science , El- Shorouk Academy , Cairo, Egypt

Abstract

This study introduces an enhanced computational framework for solving differential equations (DEs) by leveraging a Modified Variational Iteration Method (MVIM). The method is applied to derive integral transforms such as Laplace, Sumudu, and Elzaki for linear and nonlinear DEs with constant coefficients. MVIM, combining iterative refinements with variational techniques, demonstrates robust accuracy and efficiency, reducing computational complexity while preserving precision. To validate the efficacy of the proposed approach, examples are provided alongside comparisons with classical methods. The results highlight its potential applications in mathematical physics, engineering models, and dynamic systems analysis, offering a streamlined approach for solving DEs and deriving integral transforms.The application of MVIM for computing integral transforms, such as the Laplace, Sumudu, Elzaki, and Natural transforms, has been demonstrated to be highly effective. The method’s ability to directly compute these transforms from the differential equation solutions eliminates the need for intermediate steps, making it an appealing choice for practical applications in physics, engineering, and applied mathematics.

Several examples and case studies were presented to validate the method’s efficiency and reliability. Our findings show that MVIM is particularly advantageous when dealing with complex boundary conditions, nonlinearities, and systems with fractional derivatives

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