Stability analysis and a numerical scheme for fractional Klein‐Gordon equations

Abstract

Fractional order nonlinear Klein‐Gordon equations (KGEs) have been widely studied in the fields like; nonlinear optics, solid state physics, and quantum field theory. In this article, with help of the Sumudu decomposition method (SDM), a numerical scheme is developed for the solution of fractional order nonlinear KGEs involving the Caputo's fractional derivative. The coupled method provides us very efficient numerical scheme in terms of convergent series. The iterative scheme is applied to illustrative examples for the demonstration and applications.

Ulam–Hyers–Mittag-Leffler stability for ψ-Hilfer fractional-order delay differential equations

“…In [22], results for a delay differential equation were obtained using the Picard operator method, and in [23] the authors adopted a similar approach to establish the existence and uniqueness results for a Caputo-type fractional-order delay differential equation. In [31,32], the authors gave stability and numerical schemes for two classes of fractional equations. Sousa and Oliveira [33] proposed the ψ-Hilfer fractional differentiation operator and established ψ-Hilfer fractional differential equations.…”

Application of the modified operational matrices in multiterm variable‐order time‐fractional partial differential equations

“…where u(x, t) is the unknown function, f 0 (x), f 1 (x), 0 (t), 1 (t) are known functions defined over the interval = [0, 1] × [0, ∞), and F is a nonlinear function. Also, the symbol D (x,t) t denotes the variable-order Caputo fractional derivative operator with respect to variable t, which is defined by the following formula 10,[26][27][28] :…”

Construction of ( n + 1 ) $(n+1)$ -dimensional dual-mode nonlinear equations: multiple shock wave solutions for ( 3 + 1 ) $(3+1)$ -dimensional dual-mode Gardner-type and KdV-type

“…. As future work, we may consider a fractional version of dual-mode equations and conduct the same analysis as that used in [42][43][44][45][46][47].…”