An analysis for Klein–Gordon equation using fractional derivative having Mittag–Leffler‐type kernel

Abstract

Within this paper, we present an analysis of the fractional model of the Klein–Gordon (K‐G) equation. K‐G equation is considered as one of the significant equations in mathematical physics that describe the interaction of soliton in a collision less plasma. In a novel aspect of this work, we have used the latest form of fractional derivative (FCs), which contains the Mittag–Leffler type of kernel. The homotopy analysis transform method (HATM) is being taken to solve the fractional model of the K‐G equation. A convergence study of HATM has been studied. The existence and uniqueness of the solution for the fractional K‐G equation are presented. For verifying the obtained numerical outcomes regarding accuracy and competency, we have given different graphical presentations. Figures are reflecting that a novel form of the technique is a good organization in respect of proficiency and accurateness to solve the mentioned fractional problem.

On a system of q‐modified Laplace transform and its applications

“…An analysis of the fractional model has been given by Kumar and Baleanu. 57 The action of the fractional Laplace Transforms TABLE 1 q-modified Laplace transform of some preliminary functions S. no. function q-modified Laplace transform…”

“…The fractional calculus also finds applications in different fields of science, including the theory of fractals, numerical analysis, physics, engineering, biology, economics, and finance. An analysis of the fractional model has been given by Kumar and Baleanu 57 . The action of the fractional Laplace Transforms introduced by Medina et al 58 on the fractional derivative of the Riemann‐Liouville fractional operator.…”

A distributed‐order fractional diffusion equation with a singular density function: Analysis and approximation

“…Classical Fickian diffusion partial differential equation (PDE) governs the scaling limit of the random walk where the underlying particle jumps have a finite variance, which leads to a normal diffusion characterized by a linear growth of the mean‐squared displacement (MSD) in time $$$ \left\langle \boldsymbol{x}{(t)}^2\right\rangle \simeq t $$$ 1 . In many scenarios, for example, the transport of solutes in heterogeneous porous media, the diffusion is anomalous and is characterized by a power‐law growth of the MSD in time $$$ \left\langle \boldsymbol{x}{(t)}^2\right\rangle \simeq {t}^{\alpha } $$$ for $$$ 0<\alpha <1 $$$, which corresponds to the subdiffusion processes modeled by the following time‐fractional PDE (FPDE) $$$ {\partial}_t^{\alpha }u-\Delta u=f $$$ 1–11 . Here, $$$ {\partial}_t^{\alpha } $$$ is the Caputo fractional differential operator defined by 12 $$$...$$…”

“…Classical Fickian diffusion partial differential equation (PDE) governs the scaling limit of the random walk where the underlying particle jumps have a finite variance, which leads to a normal diffusion characterized by a linear growth of the mean-squared displacement (MSD) in time ⟨x(t) 2 ⟩ ≃ t. 1 In many scenarios, for example, the transport of solutes in heterogeneous porous media, the diffusion is anomalous and is characterized by a power-law growth of the MSD in time ⟨x(t) 2 ⟩ ≃ t 𝛼 for 0 < 𝛼 < 1, which corresponds to the subdiffusion processes modeled by the following time-fractional PDE (FPDE) 𝜕 𝛼 t u − Δu = 𝑓 . [1][2][3][4][5][6][7][8][9][10][11] Here, 𝜕 𝛼 t is the Caputo fractional differential operator defined by 12 0…”

Approximation of the Time-Fractional Klein-Gordon Equation using the Integral and Projected Differential Transform Methods

“…Numerous investigations have been done on this problem e.g., Golmankhaneh and Baleanu (2011) used a Homotopy perturbation method to find exact solutions of the nonlinear fractional K-G equation. Also, Saelao and Yokchoo (2020) used the Adomian decomposition method to solve the K-G equation whereas Kumar and Baleanu (2021) used the homotopy analysis transform method with the fractional-derivative of the Mittag-Leffler type of kernel. In Saifullah et al (2022), the general series solution of the nonlinear time-fractional Klien Gordan equation with power law kernel is established by the composition of double Laplace transform with the decomposition method.…”