title: THE FUNDAMENTAL SOLUTIONS FOR MULTI-TERM MODIFIED POWER LAW WAVE EQUATIONS IN A FINITE DOMAIN.
abstract: Fractional partial differential equations with more than one fractional derivative term in time, such as the Szabo wave equation, or the power law wave equation, describe important physical phenomena. However, studies of these multi-term time-space or time fractional wave equations are still under development. In this paper, multi-term modified power law wave equations in a finite domain are considered. The multi-term time fractional derivatives are defined in the Caputo sense, whose orders belong to the intervals (1, 2], [2, 3), [2, 4) or (0,n) (n> 2), respectively. Analytical solutions of the multi-term modified power law wave equations are derived. These new techniques are based on Luchko's Theorem, a spectral representation of the Laplacian operator, a method of separating variables and fractional derivative techniques. Then these general methods are applied to the special cases of the Szabo wave equation and the power law wave equation. These methods and techniques can also be extended to other kinds of the multi-term time-space fractional models including fractional Laplacian.
authors.affiliation: Department of Mathematical, Qinghai Normal University, Xining 810008, China.
authors.forename: H
authors.initials: H
authors.lastname: Jiang
authors.affiliation: Mathematical Sciences, Queensland University of Technology, GPO Box 2434, Brisbane, Qld. 4001, Australia.
authors.forename: F
authors.initials: F
authors.lastname: Liu
authors.affiliation: Department of Statistics and Probability, Michigan State University, East Lansing, Michigan 48824.
authors.forename: M M
authors.initials: MM
authors.lastname: Meerschaert
authors.affiliation: Department of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824.
authors.forename: R J
authors.initials: RJ
authors.lastname: McGough
keywords: Analytical solutions; Dirichlet boundary conditions; Szabo wave equation; power law wave equation; the multi-term time-space fractional wave equations