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<p>This paper investigates the application of the spectral tau collocation method and the power series method to derive new spectral solutions of the fractional-order Riccati differential equation. We propose an approach that utilises the operational matrix of shifted Legendre polynomials, constructed based on the orthogonality property of polynomials and the Caputo fractional derivative, to efficiently address nonlinear fractional problems. By transforming the fractional differential equation (FDE) into a system of nonlinear algebraic equations and applying an initial guess within Newton’s iterative scheme, we determine the polynomial coefficients that yield the approximate solution. In addition, the power series method is employed to solve nonlinear FDE of order <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mu \in (0,1)\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mo>∈</mo> <mo stretchy="false">(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>. The results demonstrate that the solutions obtained via the power series method are in strong agreement with the exact solutions, thereby confirming the accuracy of the approach. Furthermore, we investigate the existence and uniqueness of solutions to the Riccati FDE with both constant and variable coefficients and establish the convergence of the power series method. Illustrative examples are provided to validate the effectiveness and reliability of the proposed methodologies.</p>

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An investigation on fractional Riccati differential equation with variable coefficient—revisited

  • 期刊论文

  • 发表日期：

    2026年2月24日

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  • Sumit Kumar,
  • Sunil Kumar,
  • Shaher Momani

摘要

This paper investigates the application of the spectral tau collocation method and the power series method to derive new spectral solutions of the fractional-order Riccati differential equation. We propose an approach that utilises the operational matrix of shifted Legendre polynomials, constructed based on the orthogonality property of polynomials and the Caputo fractional derivative, to efficiently address nonlinear fractional problems. By transforming the fractional differential equation (FDE) into a system of nonlinear algebraic equations and applying an initial guess within Newton’s iterative scheme, we determine the polynomial coefficients that yield the approximate solution. In addition, the power series method is employed to solve nonlinear FDE of order \(\mu \in (0,1)\) μ ∈ ( 0 , 1 ) . The results demonstrate that the solutions obtained via the power series method are in strong agreement with the exact solutions, thereby confirming the accuracy of the approach. Furthermore, we investigate the existence and uniqueness of solutions to the Riccati FDE with both constant and variable coefficients and establish the convergence of the power series method. Illustrative examples are provided to validate the effectiveness and reliability of the proposed methodologies.