Numerical analysis of multi-dimensional time-fractional diffusion problems under the Atangana-Baleanu Caputo...

Area of Science:

• Numerical analysis
• Fractional calculus
• Partial differential equations

Background:

• Fractional diffusion equations model complex anomalous diffusion processes.
• Accurate analytical and numerical solutions are crucial for understanding these phenomena.
• The Atangana-Baleanu derivative in the Caputo sense offers a robust framework for fractional modeling.

Purpose of the Study:

• To introduce and validate the Elzaki homotopy perturbation transform scheme (EHPTS) for multi-dimensional fractional diffusion equations.
• To demonstrate the efficiency and accuracy of EHPTS in approximating solutions.
• To analyze the convergence properties and error bounds of the proposed method.

Main Methods:

• Application of the Elzaki transform (ET) to derive a recurrence relation.
• Implementation of the homotopy perturbation scheme (HPS) using the derived relation.
• Utilizing the Atangana-Baleanu derivative in the Caputo sense for fractional order.
• Graphical representation of solutions using 2D plots and 3D surfaces.

Main Results:

• The EHPTS successfully generates a convergent series solution that approximates the exact solution.
• Graphical representations illustrate the behavior of the approximate solutions.
• Error analysis confirms the high accuracy of the EHPTS, with solutions closely matching exact values.
• The method is shown to be straightforward and efficient.

Conclusions:

• The Elzaki homotopy perturbation transform scheme (EHPTS) is a powerful and efficient tool for solving multi-dimensional fractional diffusion equations.
• The method offers a simple and reliable approach for obtaining accurate approximate solutions.
• EHPTS demonstrates significant potential for application to a broader range of fractional derivative problems.