Study of Some Analytical Methods for Solving Fredholm Integral Equation of the Second Kind
Keywords:
Integral Equations, Fredholm Equation, Separable Kernel, Adomian Method, Successive ApproximationsAbstract
Integral equations are considered among the most important mathematical tools in modeling physical and engineering phenomena, especially Fredholm integral equations of the second kind, which arise in problems such as heat diffusion and quantum mechanics. This research aims to study the concept of integral equations and classify them according to the type of kernel, as well as to review various analytical methods for solving Fredholm integral equations of the second kind. In addition, numerical applications using MATLAB are presented to verify the validity of the solutions.
Four main analytical methods are employed: the Adomian Decomposition Method, the Modified Method, the Direct Computation Method for separable kernels, and the Successive Approximations Method. These methods are applied to selected problems, and MATLAB codes are developed for each method to compare analytical and numerical solutions.
The results show complete agreement between the analytical solutions and the numerical results obtained using MATLAB for all selected examples. The analytical methods also confirm the existence and uniqueness of the solution under appropriate conditions. Furthermore, the numerical applications demonstrate the convergence speed of iterative methods and the accuracy of the obtained solutions.
This study confirms that combining analytical methods with numerical applications using MATLAB provides a deeper understanding of integral equations and facilitates the verification of solution accuracy, thereby supporting their use in various scientific and engineering applications.
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