Computational Methods for Differential Equations
Volume:10 Issue: 3, Summer 2022

This paper deals with a typical Caputo fractional differential equation. This equation appears in important applications such as modeling of medicine distributed throughout the body via injection and equation for general population growth. We use the fixed point theory of concave operators in specific normed spaces to find a parameter interval for which the unique positive solution exists. Some properties of positive solutions are studied and illustrative examples are given.

In this paper, the transport of flow and heat transfer through parallel plates arranged horizontally against each other is studied. The mechanics of fluid transport and heat transfer are formulated utilizing systems of the coupled higher-order numerical model. This governing transport model is investigated by applying the variation of the parameter’s method. Result obtained from the analytical study is reported graphically. It is observed from the generated result that the velocity profile and thermal profile drop by increasing the squeeze parameter. The drop inflow is due to limitations in velocity as plates are close to each other. Also, thermal transfer due to flow pattern causes decreasing boundary layer thickness at the thermal layer, consequently drop in thermal profile. The analytical obtained result from this study is compared with the study in literature for simplified cases, this shows good agreement. The obtained results may therefore provide useful insight to practical applications including food processing, lubrication, and polymer processing industries amongst other relevant applications.

In this manuscript, we review fractal calculus and the analogues of both local Fourier transform with its related properties and Fourier convolution theorem are proposed with proofs in fractal calculus. The fractal Dirac delta with its derivative and the fractal Fourier transform of the Dirac delta is also defined. In addition, some important applications of the local fractal Fourier transform are presented in this paper such as the fractal electric current in a simple circuit, the fractal second order ordinary differential equation, and the fractal Bernoulli-Euler beam equation. All discussed applications are closely related to the fact that, in fractal calculus, a useful local fractal derivative is a generalized local derivative in the standard calculus sense. In addition, a comparative analysis is also carried out to explain the benefits of this fractal calculus parameter on the basis of the additional alpha parameter, which is the dimension of the fractal set, such that when α = 1, we obtain the same results in the standard calculus.

In this work, we use the symmetry of the Lie group analysis as one of the powerful tools that deals with the wide class of fractional order differential equation in the Riemann-Liouville concept. We employ the classical Lie symmetries to obtain similarity reductions of nonlinear time-fractional Benjamin-Ono equation and then, we find the related exact solutions for the derived generators. Finally, according to the Lie symmetry generators obtained, we construct conservation laws for related classical vector fields of time-fractional Benjamin-Ono equation.

In this paper, we are interested in the construction of an explicit third-order stochastic Runge–Kutta (SRK3) schemes for the weak approximation of stochastic differential equations (SDEs) with the general diffusion coefficient b(t, x). To this aim, we use the Itˆo-Taylor method and compare them with the stochastic expansion of the approximation. In this way, the authors encountered a large number of equations and could find to derive four families for SRK3 schemes. Also, we investigate the mean-square stability (MS-stability) properties of SRK3 schemes for a linear SDE. Finally, the proposed families are implemented on some examples to illustrate convergence results.

Based on the extragradient-like method combined with shrinking projection, we propose two algorithms, the first algorithm is obtained using sequential computation of extragradient-like method and the second algorithm is obtained using parallel computation of extragradient-like method, to find a common point of the set of fixed points of a nonexpansive mapping and the solution set of the equilibrium problem of a bifunction given as a sum of the finite number of H ̈older continuous bifunctions. The convergence theorems for iterative sequences generated by the algorithms are established under widely used assumptions for the bifunction and its summands

Among the various causes of heroin addiction, the use of prescription opioids is one of the main reasons. In this article, we introduce and analyze a two-strain epidemic model with the superinfection for modeling the effect of prescribed opioids abuse on heroin addiction. Our model contains the impact of relapse of individuals under treatment/rehabilitation to drug abuse in each strain. We extract the basic reproductive ratio, the invasion numbers and study the occurrence of backward bifurcation in strain dominance equilibria, i.e., boundary equilibria. Also, we explore both the local and global stability of DFE and boundary equilibria under suitable conditions. Furthermore, we study the existence of the coexistence equilibrium point. We prove that when R0 < 1, the coexistence equilibrium point can exist, i.e., backward bifurcation occurs in coexistence equilibria. Finally, we use numerical simulation to describe the obtained analytical results.

Recently, finding exact solutions of nonlinear fractional differential equations has attracted great interest. In this work, the space time-fractional Klein-Gordon equation with cubic nonlinearities is examined. Firstly, suitable exact soliton solutions are formally extracted by using the solitary wave ansatz method. Some solutions are also illustrated by the computer simulations. Besides, the modified Kudryashov method is used to construct exact solutions of this equation.

In this article, an efficient method for approximating the solution of the generalized Burgers-Huxley (gB-H) equation using a multiquadric quasi-interpolation approach is considered. This method consists of two phases. First, the spatial derivatives are evaluated by MQ quasi-interpolation, So the gB-H equation is reduced to a nonlinear system of ordinary differential equations. In phase two, the obtained system is solved by using ODE solvers. Numerical examples demonstrate the validity and applicability of the method.

In the present study, we investigate the conformable space-time fractional cubic-quartic nonlinear Schrodinger equation with three different laws of nonlinearity namely, parabolic law, quadratic-cubic law, and weak non-local law. This model governs the propagation of solitons through nonlinear optical fibers. An effective approach namely, the exp(−Φ(ξ)) expansion method is applied to construct some new soliton solutions of the governing model. Consequently, the dark, singular, rational and periodic solitary wave solutions are successfully revealed. The comparisons with other results are also presented. In addition, the dynamical structures of obtained solutions are presented through 3D and 2D plots.

A novel local meshless scheme based on the radial basis function (RBF) is introduced in this article for price multi-asset options of even European and American types based on the Black-Scholes model. The proposed approach is obtained by using operator splitting and repeating the schemes of Richardson extrapolation in the time direction and coupling the RBF technology with a finite-difference (FD) method that leads to extremely sparse matrices in the spatial direction. Therefore, it is free of the ill-conditioned difficulties that are typical of the standard RBF approximation. We have used a strong iterative idea named the stabilized Bi-conjugate gradient process (BiCGSTAB) to solve highly sparse systems raised by the new approach. Moreover, based on a review performed in the current study, the presented scheme is unconditionally stable in the case of independent assets when spatial discretization nodes are equispaced. As seen in numerical experiments, it has a low computational cost and generates higher accuracy. Finally, the proposed local RBF scheme is very versatile so that it can be used easily for solving numerous models and obstacles not just in the finance sector, as well as in other fields of engineering and science.

Prabhakar fractional operator was applied recently for studying the dynamics of complex systems from several branches of sciences and engineering. In this manuscript, we discuss the regularized Prabhakar derivative applied to fractional partial differential equations using the Sumudu homotopy analysis method(PSHAM). Three illustrative examples are investigated to confirm our main results.

The rapid spread of coronavirus disease (COVID-19) has increased the attention to the mathematical modeling of spreading the disease in the world. The behavior of spreading is not deterministic in the last year. The purpose of this paper is to present a stochastic differential equation for modeling the data sets of the COVID-19 involving infected, recovered, and dead cases. At first, the time series of the covid-19 is modeled with the Ornstein-Uhlenbeck process and then using the Ito lemma and Euler approximation the analytical and numerical simulations for the stochastic differential equations are achieved. Parameters estimation is done using the maximum likelihood estimator. Finally, numerical simulations are performed using reported data by the world health organization for case studies of Italy and Iran. The numerical simulations and root mean square error criteria confirm the accuracy and efficiency of the findings of the present study.

In the current study, we consider the generalized Pochhammer-Chree equation with a term of order n. Based on the (1/G0)-expansion method and with the aid of symbolic computation, we construct some distinct exact solutions for this nonlinear model. Various exact solutions are produced to the studied equation including singular solutions and periodic wave solutions. In addition, 2D, 3D, and contour plots are graphed for all obtaining solutions via choosing the suitable values for the involved parameters. All gained solutions verify the governing equation.

This paper presents a numerical method for solving a class of fractional optimal control problems (FOCPs) based on numerical polynomial approximation. The fractional derivative in the dynamic system is described in the Caputo sense. We used the approach to approximate the state and control functions by the Mott polynomials (M-polynomials). We introduced the operational matrix of fractional Riemann-Liouville integration and apply it to approximate the fractional derivative of the basis. We investigated the convergence of the new method and some examples are included to demonstrate the validity and applicability of the proposed method.

This paper generated the novel approach called the Clique polynomial method (CPM) using the clique polynomials raised in graph theory. Nonlinear initial value problems are converted into nonlinear algebraic equations by discretion with suitable grid points in the current approach. We solved highly nonlinear initial value problems using the Homotopy analysis method (HAM) and Clique polynomial method (CPM). Obtained results reveal that the present technique is better than HAM that is discussed through tables and simulations. Convergence analysis is reflected in terms of theorems.

We examine the diffusion equation on the sphere. In this sense, we answer the question of the symmetry classification. We provide the algebra of symmetry and build the optimal system of Lie subalgebras. We prove for one-dimensional optimal systems of Eq. (1.4), space is expanding Ricci solitons. Reductions of similarities related to subalgebras are classified, and some exact invariant solutions of the diffusion equation on the sphere are presented.

In this study, one explicit and one implicit finite difference scheme is introduced for the numerical solution of time-fractional Riesz space diffusion equation. The time derivative is approximated by the standard Gr¨unwald Letnikov formula of order one, while the Riesz space derivative is discretized by Fourier transform-based algorithm of order four. The stability and convergence of the proposed methods are studied. It is proved that the implicit scheme is unconditionally stable, while the explicit scheme is stable conditionally. Some examples are solved to illustrate the efficiency and accuracy of the proposed methods. Numerical results confirm that the accuracy of present schemes is of order one.

In this paper, we discuss the new generalized fractional operator. This operator similarly conformable derivative satisfies properties such as the sum, product/quotient, and chain rule. Laplace transform is defined in this case, and some of its properties are stated. In the following, the Sturm-Liouville problems are investigated, and also eigenvalues and eigenfunctions are obtained.

In the present paper, a modified simple equation method is used to obtain exact solutions of the equal width wave Burgers and modified equal width wave Burgers equations. By giving specific values to the parameters, particular solutions are obtained and the plots of solutions are drawn. It shows that the proposed method can be easily generalized to solve a variety of non-linear equations by implementing a robust and straightforward algorithm without the need for any tools.