Abstract
Aiming at the initial boundary value problem of variable-order time-fractional wave equations in one-dimensional space, a numerical method using second-order central difference in space and H2N2 approximation in time is proposed. A finite difference scheme with second-order accuracy in space and 3 − γ ∗ order accuracy in time is obtained. The stability and convergence of the scheme are further discussed by using the discrete energy analysis method. A numerical example shows the effectiveness of the results.
Highlights
Due to the non-locality of fractional calculus, more and more problems in physical science, electromagnetism, electrochemistry, diffusion and general transport theory can be described by the fractional calculus approach, among which the Riemann-Liouville fractional derivative and the Caputo fractional derivative are the most widely used [1,2,3,4]
More and more researchers found that a variety of important dynamical problems exhibit fractional-order behavior that may vary with time, space, or other conditions. This phenomenon indicates that variable-order fractional calculus is a natural choice to provide an effective mathematical framework for the description of complex problems
Suppose the function f ðtÞ is defined on the interval 1⁄20, T, 1 < γðtÞ < 2, the variable-order Caputo fractional derivative is defined as 1 − γðtÞÞ
Summary
Due to the non-locality of fractional calculus, more and more problems in physical science, electromagnetism, electrochemistry, diffusion and general transport theory can be described by the fractional calculus approach, among which the Riemann-Liouville fractional derivative and the Caputo fractional derivative are the most widely used [1,2,3,4]. More and more researchers found that a variety of important dynamical problems exhibit fractional-order behavior that may vary with time, space, or other conditions. This phenomenon indicates that variable-order fractional calculus is a natural choice to provide an effective mathematical framework for the description of complex problems. : ð2Þ uðx, 0Þ = φðxÞ, utðx, 0Þ = ψðxÞ, x ∈ ð0, LÞ: ð3Þ uð0, tÞ = 0, uðL, tÞ = 0, t ∈ 1⁄20, T: ð4Þ where 1 < γðtÞ < 2,C0 Dγt ðtÞuðx, tÞ is the variable-order Caputo fractional derivative, f ðx, tÞ, φðxÞ, ψðxÞ are given suffificiently smooth functions and satisfy φð0Þ = ψð0Þ, φðLÞ = ψðLÞ.
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