Applying Discrete Homotopy Analysis Method for Solving Fractional Partial Differential Equations

Area of Science:

• Numerical Analysis
• Fractional Calculus
• Partial Differential Equations

Background:

• Fractional partial differential equations (FPDEs) model complex phenomena.
• Analytical solutions for FPDEs are often challenging to obtain.
• Numerical methods are crucial for solving FPDEs.

Purpose of the Study:

• To develop a space discrete version of the homotopy analysis method (DHAM).
• To apply DHAM for solving linear and nonlinear fractional partial differential equations.
• To analyze the convergence properties and accuracy of the DHAM.

Main Methods:

• Developed a space discrete homotopy analysis method (DHAM).
• Applied DHAM to fractional partial differential equations with time derivative order α (0 < α ≤ 1).
• Utilized an auxiliary parameter ℏ to ensure convergence of solution series.

Main Results:

• Demonstrated the efficiency and accuracy of DHAM using test problems.
• Achieved good agreement between DHAM results and exact solutions for α = 1.
• The auxiliary parameter ℏ effectively controls the convergence region.

Conclusions:

• DHAM is an efficient and accurate numerical technique for FPDEs.
• The method provides a reliable approach for analyzing fractional differential equations.
• DHAM offers a straightforward way to ensure the convergence of series solutions.