A Discretization Approach for the Nonlinear Fractional Logistic Equation
Mohammad Izadi1, Hari M Srivastava2,3,4
1Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman 76169-14111, Iran.
Entropy (Basel, Switzerland)
|December 8, 2020
Summary
This study introduces a new numerical method for solving fractional logistic differential equations, crucial for modeling biological and social systems. The developed local discontinuous Galerkin method ensures numerical stability and accuracy for these complex equations.
Area of Science:
- Computational Mathematics
- Applied Mathematics
- Mathematical Biology
Background:
- Fractional logistic differential equations model complex phenomena in biology and social sciences.
- Accurate numerical solutions are essential for understanding these models.
- Existing methods may face challenges with stability and nonlinear terms.
Purpose of the Study:
- To develop and investigate a local discontinuous Galerkin method for fractional logistic differential equations.
- To ensure the numerical stability of the proposed method.
- To assess the method's accuracy and utility compared to existing schemes.
Main Methods:
- The Liouville-Caputo definition of the fractional derivative is used.
- Upwind numerical fluxes are employed to prove numerical stability in the L∞ norm.
- Shifted Legendre polynomials reduce the weak form to algebraic equations.
- Product approximation technique handles nonlinear terms.
Main Results:
- The local discontinuous Galerkin method is developed and its numerical stability is proven.
- The method effectively reduces the fractional differential equation to a system of algebraic equations.
- Numerical calculations demonstrate the utility of the method for linear and nonlinear problems.
Conclusions:
- The developed local discontinuous Galerkin method provides a stable and accurate approach for solving fractional logistic differential equations.
- The method's effectiveness is validated through numerical experiments on various problems.
- This technique offers a valuable tool for researchers in applied mathematics and related fields.
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