THE FUNDAMENTAL SOLUTIONS FOR MULTI-TERM MODIFIED POWER LAW WAVE EQUATIONS IN A FINITE DOMAIN

Area of Science:

• * Mathematical Physics
• * Applied Mathematics
• * Fractional Calculus

Background:

• * Multi-term time-space or time fractional wave equations are crucial for modeling physical phenomena but remain under active research.
• * Existing studies often focus on single fractional derivative terms, leaving multi-term equations less explored.
• * The Szabo and power law wave equations are examples of significant multi-term fractional wave models.

Purpose of the Study:

• * To derive analytical solutions for multi-term modified power law wave equations within a finite domain.
• * To introduce and apply novel techniques for solving these complex fractional differential equations.
• * To extend these methods to specific cases like the Szabo and power law wave equations and other fractional models.

Main Methods:

• * Application of Luchko's Theorem for fractional calculus.
• * Utilization of spectral representation of the Laplacian operator.
• * Employing the method of separating variables and advanced fractional derivative techniques.
• * Defining fractional derivatives in the Caputo sense with various order intervals (1, 2], [2, 3), [2, 4), or (0, n).

Main Results:

• * Successful derivation of analytical solutions for multi-term modified power law wave equations.
• * Demonstrated applicability of the developed techniques to the Szabo and power law wave equations.
• * Provided a generalized framework for solving a class of multi-term time-space fractional models.

Conclusions:

• * The study offers effective analytical solutions for a significant class of multi-term fractional wave equations.
• * The presented methods provide a robust framework for analyzing complex fractional phenomena.
• * These techniques can be extended to a broader range of fractional partial differential equations, including those involving the fractional Laplacian.