Solution Of Nonlinear Integro-Differential Equations by Using Laplace Decomposition Method and Series Solution Method

Objectives: To investigate two mutually beneficial methods: One is the Laplace Adomian decomposition method, and the second is the series solution method. This study intends to investigate the efficiency of both approaches in resolving nonlinear integral-differential equations. Methods: Each strategy’s main ideas and theoretical underpinnings are methodically described, along with each approach’s advantages and uses. By contrasting their performances, this work sheds light on the effectiveness and practicality of the series solution method and the Laplace decomposition method. Findings: This study reveals that combining these techniques leads to a very successful approach to solving challenging nonlinear situations. Additionally, the results show that this dual strategy increases computing efficiency and improves solution correctness. Overall, the results indicate that the Laplace decomposition method and the series solution method provide a strong framework for resolving nonlinear integro-differential equations, making them an important addition to computational mathematics. Novelty: This study is unique in that it combines the Laplace decomposition method with the series solution method, resulting in a novel dual technique for solving nonlinear integro-differential equations. The integrated technique improves computing efficiency and solution accuracy. The paper substantially adds to computational mathematics by developing a strong framework for dealing with complicated nonlinear issues. Keywords: Laplace Decomposition Method; Adomian Polynomials; Series Solution Method; Nonlinear IntegroDifferential Equations; Analytical Solution