Semi analytical solution strategy for fractional Fornberg Whitham equation using Temimi Ansari method

Area of Science:

• Nonlinear Partial Differential Equations
• Fractional Calculus
• Mathematical Physics

Background:

• The Fornberg-Whitham equation models phenomena like shallow water waves.
• Fractional calculus offers advanced modeling capabilities for systems with memory.
• Comparing fractional derivative definitions is crucial for accurate physical interpretations.

Purpose of the Study:

• To derive semi-analytical solutions for the fractional Fornberg-Whitham equation.
• To analyze the impact of different fractional derivative definitions (Atangana-Baleanu-Caputo and Caputo) on solutions.
• To rigorously examine the existence, uniqueness, and convergence of the proposed solutions.

Main Methods:

• Application of the Fractional Temimi-Ansari Method (FTAM).
• Definition of fractional derivatives in the Atangana-Baleanu-Caputo (ABC) and Caputo senses.
• Convergence analysis of the FTAM framework.
• Validation against exact solutions for specific fractional orders.

Main Results:

• Semi-analytical solutions for the fractional Fornberg-Whitham equation were successfully obtained.
• The Atangana-Baleanu-Caputo derivative, with its Mittag-Leffler kernel, demonstrated enhanced memory effects.
• The study confirmed the existence and uniqueness of the derived solutions.
• FTAM solutions showed good agreement with exact solutions.

Conclusions:

• The FTAM is an effective technique for solving fractional differential equations.
• The ABC derivative definition is superior in capturing long-range dependencies and memory effects.
• Understanding the nuances of fractional derivatives is vital for accurate modeling in physics and engineering.