Solving Fredholm Integro-Differential Equations of Mixed-Order by α-Dickson Matrix-Collocation Method
- 1Department of Mathematics, College of Science, University of Sulaimani, Sulaymaniyah, Iraq
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.
Related
- Pell–Lucas Collocation Method for Time-Fractional Partial Integro-Differential Equations
- An optimal order fractional backward collocation method for adjoint Volterra integro-differential equations
- Two-Parameter General-Shift Lucas Polynomial Spectral Collocation for Time-Fractional Nonlinear Partial Integro-Differential Equations
- Piecewise B-Spline Collocation Methods for Multi-Term Nonlinear Caputo-Type Volterra Integro-Differential Systems
- Solving Fredholm Integral Equations with Physics-Informed Neural Networks: Single-and double-network approaches
BibTeX
@article{ahmed2026solving,
title = {{Solving Fredholm Integro-Differential Equations of Mixed-Order by α-Dickson Matrix-Collocation Method}},
author = {Shazad Shawki Ahmed and Kurdistan Hama Ali Hama},
journal = {Tikrit Journal of Pure Science},
year = {2026},
month = oct,
doi = {10.25130/tjps.v31i5.1925},
url = {https://doi.org/10.25130/tjps.v31i5.1925},
note = {CC BY},
}
RIS
TY - JOUR
TI - Solving Fredholm Integro-Differential Equations of Mixed-Order by α-Dickson Matrix-Collocation Method
AU - Shazad Shawki Ahmed
AU - Kurdistan Hama Ali Hama
PY - 2026
DA - 2026/10/25
JO - Tikrit Journal of Pure Science
DO - 10.25130/tjps.v31i5.1925
UR - https://doi.org/10.25130/tjps.v31i5.1925
AB - 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.
SN - 1813-1662
SN - 2415-1726
ER -