Modeling and Solution of Reaction-Diffusion Equations by Using the Quadrature and Singular Convolution Methods

Area of Science:

• Numerical analysis
• Computational mathematics
• Mathematical modeling

Background:

• Reaction-diffusion equations are fundamental in modeling various natural phenomena.
• Existing numerical methods may lack efficiency or accuracy for complex models.
• Accurate solutions are crucial for understanding processes like pattern formation and disease spread.

Purpose of the Study:

• To introduce and evaluate novel numerical techniques for solving reaction-diffusion equations.
• To compare the efficiency and accuracy of polynomial, discrete singular convolution (DSC), and sinc quadrature methods.
• To analyze the influence of diffusion and reaction parameters on model solutions.

Main Methods:

• Application of polynomial, DSC, and sinc quadrature techniques to reduce differential equations.
• Utilizing the Runge-Kutta fourth-order method for solving resulting nonlinear ordinary differential equations.
• Implementation using MATLAB and comparison with existing numerical approaches.

Main Results:

• The DSC method, particularly with a regularized Shannon kernel, achieved high accuracy (error rate ≤ 10⁻⁶).
• The presented methods demonstrated ease of implementation and computational efficiency.
• Parametric analysis revealed the impact of diffusion and reaction parameters on the solutions.

Conclusions:

• The novel numerical techniques, especially DSC, provide accurate and efficient solutions for reaction-diffusion models.
• These methods offer a reliable alternative to existing techniques for complex biological and physical systems.
• Further investigation into parameter influence enhances the predictive power of these models.