Uniformly convergent computational method for singularly perturbed unsteady burger-huxley equation

Area of Science:

• Computational Mathematics
• Numerical Analysis
• Applied Mathematics

Background:

• Singularly perturbed problems, especially unsteady non-linear Burger-Huxley equations, pose challenges for classical numerical methods due to oscillations on uniform meshes.
• The presence of both singular perturbation parameters and non-linearity complicates direct numerical solutions, necessitating specialized techniques.

Purpose of the Study:

• To develop a uniformly convergent computational scheme for the unsteady non-linear Burger-Huxley problem.
• To address the limitations of traditional numerical methods in providing oscillation-free solutions for this class of problems.

Main Methods:

• Linearization of the non-linear problem using the Newton-Raphson-Kantorovich quasilinearization technique.
• Semi-discretization in time via the implicit Euler method, resulting in a system of ordinary differential equations.
• Solving the spatial ordinary differential equations using a fitted exponential cubic spline method.

Main Results:

• The proposed scheme demonstrates stability and uniform convergence.
• Achieved first-order accuracy in the temporal direction and second-order accuracy in the spatial direction.
• Numerical results confirm superior accuracy compared to existing methods for test examples.

Conclusions:

• The developed computational scheme effectively handles the complexities of the singularly perturbed unsteady non-linear Burger-Huxley problem.
• The method provides accurate and oscillation-free solutions, validating its applicability and superiority over other numerical approaches.