Abstract
In this work, we propose a novel analytical solution approach for solving a general homogeneous time-invariant fractional initial value problem in the normal form \t\t\tDtα[u(x‾,t)]=F(u(x‾,t)),0≤t<R,u(x‾,0)=f(x‾),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document} $$\\begin{aligned}& D_{t}^{\\alpha } \\bigl[u(\\overline{x},t) \\bigr] = F \\bigl(u( \\overline{x},t) \\bigr),\\quad 0\\leq t < R, \\\\& u(\\overline{x},0) = f(\\overline{x}), \\end{aligned}$$ \\end{document} where D_{t}^{alpha } is the Caputo fractional operator with 0<alpha leq 1. The solution is given analytically in the form of a convergent multi-fractional power series without using any particular treatments for the nonlinear terms. The new approach is taken to search patterns for compacton solutions of several nonlinear time-fractional dispersive equations, namely K_{alpha }(2,2), ZK_{alpha }(2,2), DD_{alpha }(1,2,2), and K_{alpha }(2,2,1). Remarkably, the graphical analysis showed that the n-term approximate memory solutions, labeled by the memory parameter 0<alpha leq 1, are continuously homotopic as they reflect, in some sense, some memory and heredity properties.
Highlights
The importance of fractional differential equations (FDEs) is stimulated by the appearance of many scientific models that have a nonlocal dynamical property
It has been observed that the universal electromagnetic, acoustic, and mechanical responses are influenced by a remnant memory which can be accurately modeled by the fractional diffusion wave equations [1]
In the Caputo case, the derivative of a constant function is zero and one can properly define the initial conditions for the fractional differential equations which can be handled by using an analogy with the classical integer-order case
Summary
The importance of fractional differential equations (FDEs) is stimulated by the appearance of many scientific models that have a nonlocal dynamical property. We have successfully applied the present approach to finding closed-form solutions for various nonlinear time-fractional dispersive equations, namely Kα(2, 2), ZKα(2, 2), DDα(1, 2, 2), and Kα(2, 2, 1) equations. In the Caputo case, the derivative of a constant function is zero and one can properly define the initial conditions for the fractional differential equations which can be handled by using an analogy with the classical integer-order case.
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