Analytical Solutions of Fractional-Order Diffusion Equations by Natural Transform Decomposition Method.

Area of Science:

• Applied Mathematics
• Numerical Analysis
• Fractional Calculus

Background:

• Fractional-order diffusion equations model complex phenomena across various scientific disciplines.
• Existing analytical techniques often involve extensive computations and may lack convergence.
• Efficient and accurate methods are needed for solving these challenging equations.

Purpose of the Study:

• To introduce and evaluate the Natural transform decomposition method (NTDM) for solving fractional-order diffusion equations.
• To demonstrate the method's applicability and efficiency through numerical examples.
• To compare NTDM with existing analytical techniques.

Main Methods:

• The Natural transform decomposition method (NTDM) is employed to derive series solutions.
• The method combines the Natural transform with the Adomian decomposition method.
• Numerical examples are presented to validate the procedure and assess performance.

Main Results:

• Series form solutions were successfully obtained for fractional-order diffusion equations.
• NTDM demonstrated a reduced computational load compared to other analytical methods.
• The method exhibited a high rate of convergence.

Conclusions:

• The Natural transform decomposition method is a powerful and efficient technique for solving fractional-order diffusion equations.
• NTDM offers advantages in terms of computational simplicity and convergence speed.
• The method's versatility allows for its application to other linear and non-linear fractional partial differential equations.