Numerical Solutions of Fractional Integro-Differential Equations with Weakly Singular Kernel by Using a Hybrid of Block-Pulse Functions and Taylor Polynomials
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Abstract
Fractional integro-differential equations (FIDEs) with weakly singular kernels arise in mathematical models of viscoelasticity, anomalous diffusion, and hereditary systems, yet closed-form solutions are rarely attainable. This study proposes a numerical scheme based on a hybrid combination of block-pulse functions and Taylor polynomials for solving both linear and nonlinear FIDEs with weakly singular kernels of the form , formulated within the Caputo fractional derivative framework. The fractional problem is transformed into an equivalent system of algebraic equations by constructing a Riemann–Liouville operational matrix and applying Newton–Cotes collocation nodes, yielding a computationally straightforward solver. Three illustrative examples were examined, including two linear equations with exact solutions and , and one nonlinear equation with an exact solution . The proposed method achieved absolute errors below across all test points using N = 1, 2 block-pulse intervals and M = 2, 3 Taylor polynomial terms. Comparison with the Legendre wavelet method (LWM) demonstrates that the present scheme matches or surpasses LWM accuracy at most collocation points while requiring fewer basis functions. These results confirm that the hybrid block-pulse–Taylor polynomial approach offers a reliable and efficient alternative for solving FIDEs encountered in applied science and engineering.
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References
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