Research Article Solution of Space-Time-Fractional Problem by Shehu Variational Iteration Method Suleyman Cetinkaya , Ali Demir , and Hulya Kodal Sevindir Department of Mathematics, Kocaeli University, Kocaeli 41380, Turkey Correspondence should be addressed to Suleyman Cetinkaya; suleyman.cetinkaya@kocaeli.edu.tr Received 27 January 2021; Revised 26 March 2021; Accepted 19 April 2021; Published 30 April 2021 Academic Editor: Marianna Ruggieri Copyright © 2021 Suleyman Cetinkaya et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. In this study, we deal with the problem of constructing semianalytical solution of mathematical problems including space-time- fractional linear and nonlinear differential equations. The method, called Shehu Variational Iteration Method (SVIM), applied in this study is a combination of Shehu transform (ST) and variational iteration method (VIM). First, ST is utilized to reduce the time-fractional differential equation with fractional derivative in Liouville-Caputo sense into an integer-order differential equation. Later, VIM is implemented to construct the solution of reduced differential equation. The convergence analysis of this method and illustrated examples confirm that the proposed method is one of best procedures to tackle space-time-fractional differential equations. 1. Introduction Last couple of decades, employing fractional differential equations in modelling of processes such as dynamical sys- tems, biology, fluid flow, signal processing, electrical net- works, reaction and diffusion procedure, and advection– diffusion–reaction process [1–4] has gained great importance since these models reflect the behaviour of the processes bet- ter than integer-order differential equations. Consequently, a great deal of methods such as [3, 4] are established to construct analytical and numerical solutions of fractional differential equations. Moreover, their existence, uniqueness, and stability have been studied by many scientists. One of the significant integral transformations is Shehu transformation proposed by Maitama and Zhao [5]. This linear transformation is a generalization of Laplace transformation. However, the Laplace transformation is obtained by substitut- ing q =1 in Shehu transformation. By this transformation, dif- ferential equations are reduced into simpler equations. Various methods such as the homotopy perturbation method (HPM) and VIM are utilized to establish approxi- mate solutions of differential equations of any kind [6, 7]. As a result, it is employed widely to deal with differential equations in various branches of science [8–11]. VIM has been modified by many researchers to improve this method. By modified VIM, the approximate solutions of initial value problems can be established by making use of an initial condition. 2. Preliminaries In this section, preliminaries, notations, and features of the frac- tional calculus are given [12, 13]. Riemann-Liouville time- fractional integral of a real valued function uðx, t Þ is defined as I α t ux, t ð Þ = 1 Γα ðÞ ð t 0 t − s ð Þ α−1 ux, s ð Þds, ð1Þ where α >0 denotes the order of the integral. The α th -order Liouville-Caputo time-fractional deriva- tive operator of uðx, t Þ is defined as ∂ α ux, t ð Þ ∂t α = I m−α t ∂ m ux, t ð Þ ∂t m   = 1 Γ m − α ð Þ ð t 0 t − y ð Þ m−α−1 ∂ m ux, y ð Þ ∂y m dy, m − 1< α < m, ∂ m ux, t ð Þ ∂t m , α = m: 8 > > > < > > > : ð2Þ Hindawi Advances in Mathematical Physics Volume 2021, Article ID 5528928, 8 pages https://doi.org/10.1155/2021/5528928