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Numerical solution of multi-dimensional time-fractional diffusion problems using an integral approach.

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|September 23, 2024
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Summary
This summary is machine-generated.

This study introduces the Mohand homotopy integral transform scheme (MHITS) for solving multi-dimensional fractional diffusion problems. The MHITS method provides accurate numerical solutions in a convergent series, matching exact results effectively.

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Area of Science:

  • Numerical Analysis
  • Fractional Calculus
  • Partial Differential Equations

Background:

  • Multi-dimensional diffusion problems involving fractional derivatives (Caputo sense) present significant numerical challenges.
  • Existing numerical methods may require specific assumptions or limitations, hindering broad applicability.
  • The need for robust and direct numerical schemes for fractional diffusion equations is critical.

Purpose of the Study:

  • To present a novel numerical scheme, the Mohand homotopy integral transform scheme (MHITS), for solving multi-dimensional fractional diffusion problems.
  • To demonstrate the direct implementation and efficiency of MHITS without requiring presumptions or hypotheses.
  • To validate the accuracy of MHITS by comparing its series solutions with exact results.

Main Methods:

  • The Mohand homotopy integral transform scheme (MHITS) is developed by combining the Mohand integral transform (MIT) and the homotopy perturbation scheme (HPS).
  • MHITS is applied directly to the recurrence relation of the fractional diffusion problems.
  • The numerical solution is obtained in the form of a convergent series.

Main Results:

  • The MHITS method successfully generates numerical solutions in a convergent series form.
  • The obtained series solutions demonstrate high accuracy and excellent agreement with the exact solutions.
  • Graphical results and error distribution plots confirm the reliability and effectiveness of the MHITS approach.

Conclusions:

  • The MHITS is a powerful and direct numerical technique for investigating multi-dimensional fractional diffusion problems.
  • The scheme offers a reliable alternative to existing methods, providing accurate results without restrictive assumptions.
  • MHITS shows significant potential for solving complex fractional differential equations in various scientific and engineering fields.