Computational Methods for Differential Equations
Volume:9 Issue: 1, Winter 2021

In this study, we investigate the first order linear fuzzy differential equations with fuzzy variable coefficients. Appearance of the multiplication of a fuzzy variable coefficient by an unknown fuzzy function in linear differential equations persuades us to employ the concept of the cross product of fuzzy numbers. Mentioned product overcomes to some difficulties we face to in the case of the usual product obtained by Zadeh’s extension principle. Under the cross product, we obtain the explicit fuzzy solutions for a fuzzy initial value problem applying the concept of the strongly generalized differentiability. Finally, some examples are given to illustrate the theoretical results. The obtained numerical results are compared with other approaches in the literature for similar parameters.

In this study, an effective collocation method based on cubic B-spline has been implemented to get the numerical solutions for the non-linear Fisher’s equation. After separating this scheme with this method, the stability of the method was proven. To check the efficiency and accuracy of the proposed method, some numerical problems have been considered. The numerical results are found in good agreement with the exact solutions.

By using the Guo-Krasnoselskii’s fixed point theorem, we investigate the existence of positive solutions for a non-autonomous fractional differential equations with integral boundary conditions of fractional order α ∈ (2, 3] in an ordered Banach space. The Fredholm integral equation has an important role in this article. Some examples are presented to illustrate the efficiency of the obtained results.

In this article, the decoupled nonlinear Schrdingers equations have been considered that describe the model of dual-core fibers with group velocity mismatch, group velocity dispersion, and spatio-temporal dispersion. These equations are analyzed using two different integrations schemes, namely, extended tanh-function and sinecosine schemes. The different kind of traveling wave solutions: solitary, topological, periodic and rational, fall out as by-product of these schemes. Finally, the existence of the solutions for the constraint conditions is also shown.

In this study, the asymptotic Adomian decomposition method (AADM) is implemented to solve fractional order Riccati differential equations. The product integration method is used to solve the singular integrals, resulted from fractional derivative. Some fractional order Riccati differential equations are presented as examples to illustrate the ability and efficiency of the proposed approach. The approximate solutions of AADM are compared with the results of the Laplace Adomian Pade method (LAPM). Generalizing AADM for solving fractional Riccati differential equations by the far-field approximation indicates the novelty of the paper.

In this paper, we solve a class of fractional optimal control problems in the sense of Caputo derivative using Genocchi polynomials. At first we present some properties of these polynomials and we make the Genocchi operational matrix for Caputo fractional derivatives. Then using them, we solve the problem by converting it to a system of algebraic equations. Some examples are presented to show the efficiency and accuracy of the method.

In this paper, we construct exact families of traveling wave (periodic wave, singular wave, singular periodic wave, singular-solitary wave and shock wave) solutions of a well-known equation of nonlinear PDE, the variable coefficients combined HirotaLakshmanan-Porsezian-Daniel (Hirota-LPD) equation with the fourth nonlinearity, which describes an important development, and application of soliton dispersion management experiment in nonlinear optics is considered, and as an achievement, a series of exact traveling wave solutions for the aforementioned equation is formally extracted. This nonlinear equation is solved by using the extended trial equation method (ETEM) and the improved tan(ϕ/2)-expansion method (ITEM). Meanwhile, the mechanical features of some families are explained through offering the physical descriptions. Analytical treatment to find the nonautonomous rogue waves are investigated for the combined Hirota-LPD equation.

This paper presents a mathematical model for transmission dynamics of Zika virus by considering standard incidence type interaction for the human to human transmission. The model involves the transmission through the bite of infected Aedes mosquitoes and human to human sexual transmission. The equilibria of the proposed model are found and the basic reproduction number R0 is computed. If R0 < 1, the disease-free equilibrium point is locally asymptotically stable and it is also globally asymptotically stable under certain conditions. The analysis shows that the model exhibits the occurrence of backward bifurcation, which suggests that when R0 < 1 is not completely sufficient for eradicating the disease where the stable disease-free equilibrium co-exists with a stable endemic equilibrium. The endemic equilibrium point of the system exists and locally asymptotically stable under some restriction on parameters, whenever R0 > 1. The sensitivity analysis is performed to identify the key parameters that affect the basic reproduction number, which can be regulated to control the transmission dynamics of the Zika. Further, this model is extended to the optimal control model and to reveals the optimal control strategies we used the Pontryagin’s Maximum Principle. It has been noticed that the optimal control gives better result than without the optimal control model. Numerical simulation is presented to support our mathematical findings.

In this work, a semialgebraic mode analysis (SAMA) is proposed for investigating the convergence of a multigrid waveform relaxation method applied to the Finite Element (FE) discretization of the heat equation in two and three dimensions. This analysis for finite element methods is more involved and more general than that for Finite Difference (FD) discretizations, since mass matrix must be considered. The proposed analysis results in a very useful tool to study the behaviour of the multigrid waveform relaxation method depending on the parameters of the problem.

In this paper, we firstly study the employing of the Monte Carlo method for solving system of linear algebraic equations and then analyze on convergence of this method. We propound new results related to the convergence of the Monte Carlo method. Additionally, we introduce a new Monte Carlo algorithm with effective techniques. Finally, we compare the efficiency of new Monte Carlo algorithm with its old version in the numerical experiments.

This paper investigates the distributed controllability of nonlocal Rayleigh beam. The mathematical problem is formulated as an abstract differential equation. It is shown that a sequence of eigenfunction of nonlocal Rayleigh beam forms Riesz basis. Based on Riesz basis properties and theory of abstract differential equation, it is proved that a vibrating nonlocal Rayleigh beam is approximately controllable under suitable distributed control force while it is not exponentially stable.

In this paper, a numerical method based on polynomial approximation is presented for the Riesz fractional telegraph equation. First, a system of fractional differential equations are obtained from the telegraph equation with respect to the time variable by using the method of lines. Then a new numerical algorithm, as well as its modification for solving fractional differential equations (FDEs) based on the polynomial interpolation, is proposed. The algorithms are designed to estimate to linear fractional systems. The convergence order and stability of the fractional order algorithms are proved. At the end three examples with known exact solutions are chosen. Numerical results show that accuracy of present scheme is of order O(∆t 2 ).

A new six order method developed for the approximation Fredholm integral equation of the second kind. This method is based on the quintic spline functions (QSF). In our approach, we first formulate the Quintic polynomial spline then the solution of integral equation approximated by this spline. But we need to develop the end conditions which can be associated with the quntic spline. To avoid the reduction accuracy, we formulate the end condition in such a way to obtain the band matrix and also to obtain the same order of accuracy. The convergence of the method is discussed by using matrix algebra. Finally, four test problems have been used for numerical illustration to demonstrate the practical ability of the new method.

In this paper, an efficient computational algorithm for the solution of Hamiltonian boundary value problems arising from bang-bang optimal control problems is presented. For this purpose, at first, based on the Pontryagin’s minimum principle, the first order necessary conditions of optimality are derived. Then, an indirect shooting method with control parameterization, in which the control function is replaced with a piecewise constant function with values and switching points taken as unknown parameters, is presented. Thereby, the problem is converted to the solution of the shooting equation, in which the values of the control function and the switching points as well the initial values of the costate variables are unknown parameters. The important advantages of this method are that, the obtained solution satisfies the first order optimality conditions, further the switching points can be captured accurately which is led to an accurate solution of the bangbang problem. However, solving the shooting equation is nearly impossible without a very good initial guess. So, in order to cope with the difficulty of the initial guess, a homotopic approach is combined with the presented method. Consequently, no priori assumptions are made on the optimal control structure and number of the switching points, and sensitivity to the initial guess for the unknown parameters is resolved too. Illustrative examples are included at the end and efficiency of the method is reported.

In this paper, we consider Sturm-Liouville problems on two symmetric disjoint intervals with two supplementary discontinuous conditions at an interior point. First, we investigate some spectral properties of boundary value problems, and obtain the asymptotic form of the eigenvalues and the eigenfunctions. Then, the eigenfunction expansion of Green’s function is presented and we prove the uniqueness theorems for the solution of the inverse problem, and reconstruct the Sturm-Liouville operator and the coefficients of boundary conditions using the Weyl m-function and spectral data. Also, numerical examples are presented.

‎In this paper, a type of time-fractional Fokker-Planck equation (FPE) of the OrnsteinUhlenbeck process is solved via Riemann-Liouville and Caputo derivatives. An analytical method based on symmetry operators is used for finding reduced form and exact solutions of the equation. A numerical simulation based on the M¨untz-Legendre polynomials is applied in order to find some approximated solutions of the equation.

In this paper, we propose a meshless regularization technique for solving an optimal shape design problem (OSD) which consists of constructing the optimal configuration of a conducting body subject to given boundary conditions to minimize a certain objective function. This problem also can be seen as the problem of building a support for a membrane such that its deflection is as close as possible to 1 in the subset D of the domain. We propose a numerical technique based on the combination of the method of fundamental solutions and application of the Tikhonov’s regularization method to obtain stable solution. Numerical experiments while solving several test examples are included to show the applicability of the proposed method for obtaining the satisfactory results.

This paper presents some new definitions and theorems about a system of linear uncertain differential equations. An existence and uniqueness of solutions of the system with initial condition will be proven. Also, it will be shown that the collection of solutions of the homogeneous uncertain system is a linear space.

The nonlinear variable coefficient Zakharov-Kuznetsov (Vc-ZK) equation is derived using reductive perturbation technique for ion-acoustic solitary waves in magnetized three-component dusty plasma having negatively charged dust particles, isothermal ions, and electrons. The equation is investigated for generalized symmetries using a recently proposed compatibility method. Some more general symmetries are obtained and group invariant solutions are also constructed for these symmetries. Besides this, the equation is also investigated for nontrivial local conservation laws

‎In this article, the application of discrete mollification as a regularization procedure for solving a nonlinear inverse problem in one dimensional space is considered. Illposedness is identified as one of the main characteristics of inverse problems. It is clear that if we have a noisy data, the inverse problem becomes unstable. As such, a numerical procedure based on discrete mollification and space marching method is applied to address the ill-posedness of the mentioned problem. The regularization parameter is selected by generalized cross validation (GCV) method. The numerical stability and convergence of the proposed method are investigated. Finally, some test problems, whose exact solutions are known, are solved using this method to show the efficiency.