Abstract
Abstract In this paper, a discrete orthogonal spline collocation method combining with a second-order Crank-Nicolson weighted and shifted Grünwald integral (WSGI) operator is proposed for solving time-fractional wave equations based on its equivalent partial integro-differential equations. The stability and convergence of the schemes have been strictly proved. Several numerical examples in one variable and in two space variables are given to demonstrate the theoretical analysis.
Highlights
Fractional partial di erential equations (FPDEs) have attracted more and more attention, which can be used to describe some physical and chemical phenomenon more accurately than the classical integerorder di erential equations
In this paper, a discrete orthogonal spline collocation method combining with a second-order Crank-Nicolson weighted and shifted Grünwald integral (WSGI) operator is proposed for solving timefractional wave equations based on its equivalent partial integro-di erential equations
Various kinds of numerical methods for solving fractional partial di erential equations (FPDEs) have been proposed by researchers, such as nite element method [2, 3], nite di erence method [4,5,6], meshless method [7, 8], wavelets method [9],spline collocation method [10,11,12] and so forth
Summary
Fractional partial di erential equations (FPDEs) have attracted more and more attention, which can be used to describe some physical and chemical phenomenon more accurately than the classical integerorder di erential equations. When studying universal electromagnetic responses involving the uni cation of di usion and wave propagation phenomena, there are processes that are modeled by equations with time fractional derivatives of order γ ∈ ( , ) [1]. The analytical solutions of fractional partial di erential equations are di cult to obtain, so many authors have resorted to numerical solution techniques based on convergence and stability. We consider the following two-dimensional time-fractional di usion-wave equation. Various kinds of numerical methods for solving FPDEs have been proposed by researchers, such as nite element method [2, 3], nite di erence method [4,5,6], meshless method [7, 8], wavelets method [9],spline collocation method [10,11,12] and so forth.
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