Computational Methods for Differential Equations
Volume:11 Issue: 4, Autumn 2023

This paper discusses a semi-analytic solution for the volatile oil influx into the well on the base of Forchheimer flow law. The solution is developed employing the concept of binary model for the two-phase petroleum hydrocarbon system in view of phase transformations and interphase mass transfer. Algorithms are developed for calculating the volatile oil reservoir key performance indicators by applying the material balance equations, which take into account the compaction behavior of rocks. A computer simulator for the volatile oil reservoir is modeled, proceeding from these algorithms. The inertial effects on the development process of a volatile oil reservoir, the rocks of which are exposed to elastic deformation, are studied by this simulator. In regard thereto, the reservoir development process is simulated in two variants in conformity with the constant depression: in the first case, it is assumed that the filtration occurs according to Darcy's law, while in the second one, the process is considered on the base of Forchheimer equation. A comparison of the results of these options made it possible to demonstrate the nature of the inertial effects on the volatile oil reservoir key performance indicators.

In this paper, we propose an explicit split-step truncated Milstein method for stochastic differential equations (SDEs) with commutative noise. We discuss the mean-square convergence properties of the new method for numerical solutions of a class of highly nonlinear SDEs in a finite time interval. As a result, we show that the strong convergence rate of the new method can be arbitrarily close to one under some additional conditions. Finally, we use an illustrative example to highlight the advantages of our new findings in terms of both stability and accuracy compared to the results in Guo et al. (2018).

An analytical study is carried out to obtain the approximate solution for the Magnetohydrodynamic (MHD) flow issue of Darcy-Forchheimer nanofluid containing motile microorganisms having viscous dissipation effect through a non-linear extended sheet employing a new approximate analytical method namely Ananthaswamy-Sivasankari Method (ASM) and also Modified Homotopy Analysis method (MHAM). The derived analytical solution is given in explicit form and is compared with the numerical solution. The graphical results are interlined to reflect the effects of various physical parameters involved in the problem. The numerical computation of the Nusselt number, the local skin friction parameter, and the Sherwood number are compared and shown in the table. Faster convergence is acquired using this strategy. The solution obtained by this method is closer to the exact solution. Also, the solution is in the simplest and most explicit form. It is applicable for all initial and boundary value problems with non-zero boundary conditions. This method can be easily extended to solve other non-linear higher order boundary value problems in physical, chemical, and biological sciences.

This paper focuses on the numerical solution of the time-fractional telegraph equation in Caputo sense with $1 < beta < 2$. The time-fractional telegraph equation models neutron transport inside the core of a nuclear reactor. The proposed numerical solution consists of two stages. First, the time-discretized scheme of this equation is obtained by the Crank-Nicolson method. The stability and convergence of results from the semi-discretized scheme are presented. In the second stage, the numerical approximation of the unknown function at specific points is achieved through the collocation method using the moving least square method. The numerical experiments analyze the impact of some parameters of the proposed method.

In the recent literature, many researchers are interested to apply standard computational methods for exact or numerical solutions of many classical nonlinear partial differential equations. Some leading methods are based on Lie group analysis, Painleve Analysis, G0/G expansion techniques, homotopy perturbation methods, and so on. The equations include complicated Navier-Stokes equation, Schrodinger equation, KdV-like equations, and so on. As a result, the glory of nonlinear dynamics can be witnessed through its applications in many fields namely: ocean engineering, plasma physics, optical communications, fluid dynamics, and much more. One of the significant observations is that whatever may be the order of nonlinear PDE, as far as the soliton and multisolitons of KdV like equation or Boussinesq equation are concerned Hirota’s method and tanh − coth method play a crucial role. The main result of the paper demonstrates that the above novel theme works well with the generalized Boussinesq equation of 10th order. In this paper, the Boussinesq equation of order ten is derived and its multi-soliton solutions are deduced by the Hirota’s method. The one soliton solution is reconfirmed using the tanh method.

This paper presents an extrapolated triangular splitting method (texttt{ETrnSplit}) to find the formal solution for the interval system of linear algebraic equations in which this method uses Kaucher interval arithmetic. Some numerical experiments are given to show the efficiency of this method.

Multiquadric radial basis functions combined with compact discretization to estimate solutions of two dimensions nonlinear elliptic type partial differential equations are presented. The scattered grid network with continuously varying step sizes helps tune the solution accuracies depending upon the location of high oscillation. The radial basis functions employing a nine-point grid network are used to improve the functional evaluations by compact formulation, and it saves memory space and computing time. A detailed description of convergence theory is presented to estimate the error bounds. The analysis is based on a strongly connected graph of the Jacobian matrix, and their monotonicity occurred in the scheme. It is shown that the present strategy improves the approximate solution values for the elliptic equations exhibiting a sharp changing character in a thin zone. Numerical simulations for the convection-diffusion equation, Graetz-Nusselt equation, Schr¨odinger equation, Burgers equation, and Gelfand-Bratu equation are reported to illustrate the utility of the new algorithm.

A numerical technique for the solution of the one-phase Stefan problem for the non-classical heat equation with a convective condition is discussed. This approach is based on a scheme introduced in [16]. The compatibility and convergence of the method is proven. Numerical examples round out the discussion.

In this paper, we study the existence, uniqueness, and finite-time stability results for fractional delayed Newton cooling law equation involving Ψ-Caputo fractional derivatives of order α ∈ (0, 1). By using Banach fixed point theorem, Henry Gronwall type retarded integral inequalities, and some techniques of Ψ-Caputo fractional calculus, we establish the existence and uniqueness of solutions for our proposed model. Based on the heat transfer model, a new criterion for finite time stability and some estimated results of solutions with time delay are derived. In addition, we give some specific examples with graphs and numerical experiments to illustrate the obtained results. More importantly, the comparison of model predictions versus experimental data, classical model, and non-delayed model shows the effectiveness of our proposed model with a reasonable precision.

In this paper, we study the algebraic structure of differential invariants of a fifth-order KdV equation, known as the Kawahara KdV equation. Using the moving frames method, we locate a finite generating set of differential invariants, recurrence relations, and syzygies among the differential invariants generators of the equation. We prove that the differential invariant algebra of the equation can be generated by two first-order differential invariants.

In this paper, the eigenvalues and corresponding eigenfunctions of a fractional order Sturm-Liouville problem (FSLP) are approximated by using the fractional differential transform method (FDTM), which is a generalization of the differential transform method (DTM). FDTM reduces the proposed fourth-order FSLP to a system of algebraic equations. The resulting coefficient matrix defines a characteristic polynomial which its roots correspond to the eigenvalues of FSLP. The obtained numerical results which are compared with the results of other papers confirm the efficiency of the method.

A class of singularly perturbed mixed type boundary value problems is considered here in this work. The domain is partitioned into two subdomains. Convection-diffusion and reaction-diffusion problems are posed on the first and second subdomain, respectively. To approximate the problem, a hybrid scheme which consists of a  second-order central difference scheme and a midpoint upwind scheme is constructed on Shishkin-type meshes. We have shown that the proposed scheme is second-order convergent in the maximum norm which is independent of the perturbation parameter. Numerical results are illustrated to support the theoretical findings.

In this paper, we describe a spectral Tau approach for approximating the solutions of a system of multi-order fractional differential equations which resulted from coronavirus disease mathematical modeling (COVID-19). The non-singular fractional derivative with a Mittag-Leffler kernel serves as the foundation for the fractional derivatives. Also, the operational matrix of fractional differentiation on the domain [0, a] is presented. Then, the convergence analysis of the proposed approximate approach is established and the error bounds are determined in a weighted L2 norm. Finally, by applying the Tau method, some of the important parameters in the model’s impact on the dynamics of the disease are graphically displayed for various values of the non-integer order of the ABC-derivative.

In this study, we aim to contribute to the increasing interest in functional differential equations by obtaining new existence theorems for non-oscillatory solutions of second-order neutral differential equations involving positive and negative terms which have not been performed in previous studies. We consider different cases for the ranges of the neutral coefficients, by utilizing the Banach contraction mapping principle. The applicability of the results is illustrated by several examples in the last section.

In this article, a numerical method is presented to solve Abel’s equations. In the given method, the solution of the equation is found as a finite expansion of the shifted Legendre polynomials. To this end, the integral and differential parts of the equation are converted to vector-matrix representations. Therefore, the equation is converted to an algebraic system of the equations and by solving it, the solution of the equation is obtained. Further, the numerical example is given to illustrate the method’s efficiency.