Solving Fredholm Integro-Differential Equations of Mixed-Order by α-Dickson Matrix-Collocation Method
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Abstract
In this article, we utilize -Dickson polynomials to improve a method for spectral collocation that is used to numerically solve mixed classical and fractional-order integro-differential equations of the Fredholm type with variable coefficients under boundary conditions (IFDEs-CF) in the Caputo sense. The approximate solutions are assumed in the form of the truncated Dickson polynomial series. This innovative method transforms the equation with conditions into a linear algebraic system for the unknown Dickson coefficients using a matrix technique. Solving this system determines the coefficients of the assumed solution. We ultimately analyze the results with respect to the α parameter in the Dickson polynomials. The behavior of the solutions is monitored in terms of the -Dickson coefficients depending on different inputs of . . This approach is computationally attractive, and we provide illustrative examples with usage explanations. Additionally, we provide specific examples to demonstrate the method's accuracy, and we use least-squares error minimization to reduce error terms within the given domain. Finally, we develop the most common application of numerical approaches in a Python program.
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