Analytical Solutions to a Nonlinear Fredholm Integral Equation Using Laplace-Series Techniques
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Abstract
This paper presents a comprehensive analytical investigation into solving a specific nonlinear Fredholm integral equation of the second kind, expressed as u(x)=x+λ∫01xtu2(t)dt. Utilizing the Laplace-series method, we derive explicit solutions and validate their accuracy through detailed mathematical procedures. The study focuses on the parameter λ , with particular emphasis on the case λ=0.7, where two distinct linear solutions emerge. We explore the derivation process, verify the solutions against special cases, and analyze their graphical representation using a MATLAB-based approach. The findings underscore the effectiveness of the Laplace-series method in addressing nonlinear integral equations and provide insights into the behavior of the solutions over the interval [0,1]. The results are further supported by numerical verification and a visual plot, offering a robust framework for understanding the equation’s solution space.
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References
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