Collocation method for solving fractional reaction-diffusion problem arising in chemistry

Area of Science:

• Numerical Analysis
• Computational Mathematics
• Fractional Calculus

Background:

• Fractional reaction-diffusion equations model complex phenomena.
• Accurate numerical solutions are crucial for understanding these systems.
• Existing methods may face computational challenges.

Purpose of the Study:

• To develop a novel numerical approach for Caputo-type fractional reaction-diffusion problems.
• To reduce the computational cost associated with solving these equations.
• To establish the validity and efficiency of the proposed method.

Main Methods:

• Utilized the collocation method with orthogonalized Bernoulli polynomial bases.
• Employed operational matrices to convert the fractional differential equation into algebraic equations.
• Analyzed convergence and error bounds for method validation.

Main Results:

• The proposed method effectively reduces the problem to a solvable system of algebraic equations.
• Operational matrices exhibit sparsity, leading to reduced computational expenses.
• Numerical examples demonstrate the method's capability and accuracy.

Conclusions:

• The developed numerical technique is efficient and reliable for solving fractional reaction-diffusion problems.
• The method offers a promising alternative to existing techniques.
• Sparsity of operational matrices is a key advantage for computational efficiency.