Computational Methods for Differential Equations
Year:2021 | Volume:9 | Issue:2|Papers Count:20

Using adaptive mesh methods is one of the strategies to improve numerical solutions in time-dependent partial di, erential equations. The moving mesh method is an adaptive mesh method, which, , rstly does not need an increase in the number of mesh points, secondly reduces the concentration of points in the steady areas of the solutions that do not need a high degree of accuracy, and , nally places the points in the areas, where a high degree of accuracy is needed. In this paper, we improved the numerical solutions for a three-phase model of avas-cular tumor growth by using the moving mesh method. The physical formulation of this model uses reaction-di, usion dynamics with the mass conservation law and appears in the format of the nonlinear system of partial di, erential equations based on the continuous density of three proliferating, quiescent, and necrotic cell catego-rizations. Our numerical results show more accurate numerical solutions, as compared to the corresponding , xed mesh method. Moreover, this method leads to the higher order of numerical convergence.

We consider a nonsmooth optimization problem with a feasible set de, ned by van-ishing constraints. First, we introduce a constraint quali, cation for the problem, named NNAMCQ. Then, NNAMCQ is applied to obtain a necessary M-stationary condition. Finally, we present a su, cient condition for M-stationarity, under gener-alized convexity assumption. Our results are formulated in terms of Mordukhovich subdi, erential.

In the present paper, the spectral meshless radial point interpolation (SMRPI) tech-nique is applied to the solution of pattern formation in nonlinear reaction-di, usion systems. Firstly, we obtain a time discrete scheme by approximating the time deriva-tive via a , nite di, erence formula, then we use the SMRPI approach to approximate the spatial derivatives. This method is based on a combination of meshless meth-ods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which act as ba-sis functions in the frame of SMRPI. In the current work, the thin plate splines (TPS) are used as the basis functions and in order to eliminate the nonlinearity, a simple predictor-corrector (P-C) scheme is performed. The e, ect of parameters and conditions are studied by considering the well known Brusselator model. Two test problems are solved and numerical simulations are reported which con, rm the e, ciency of the proposed scheme.

In this paper, we present a nonlinear parametric method to stabilize descriptor frac-tional discrete-time linear system practically. Parametric methods with the free parameters can be adjusted to obtain better performance responses like minimum norm in state feedback. The aim is assigning desirable eigenvalues to obtain sat-isfactory responses by forward state feedback and forward and propositional state feedback in new systems with large matrices. However, , nding the solution to non-linear parametric equations makes some errors. In partial eigenvalue assignment, just a part of the open-loop spectrum of the standard linear systems is reassigned, while leaving the rest of the spectrum invariant. The size of matrices, state, and input vectors are decreased and the stability is kept. At the end, summary and con-clusions are proposed and the convergence of state vectors in the descriptor fractional discrete-time system to zero is also shown by , gures in a numerical example. Our method is also compared with another method with one of orthogonality relations in our article and example.

Stochastic linear combinations of some random vectors are studied where the dis-tribution of the random vectors and the joint distribution of their coe, cients have Dirichlet distributions. A method is provided for calculating the distribution of these combinations which has been studied before. Our main result is the same as but from a di, erent point of view.

In this paper, we consider the hyperbolic Ricci-Bourguignon ow on a compact man-ifold M and show that this ow has a unique solution on short-time with imposing on initial conditions. After then, we , nd evolution equations for Riemannian cur-vature tensor, Ricci curvature tensor and scalar curvature of M under this ow. In the , nal section, we give some examples of this ow on some compact manifolds.

In this paper, we concern ourselves with the study of a class of stationary states for reaction-di, usion systems with densities having disjoint supports. Major contri-bution of this work is computing the numerical solution of problem as the rate of interaction between two di, erent species tend to in, nity. The main di, culty is the nonlinearity nature of problem. To do so, an e, cient iterative method is proposed by hybrid of the radial basis function (RBF) collocation and , nite di, erence (FD) methods to approximate the solution. Numerical results with good accuracies are achieved where the shape parameter is carefully selected. Finally, some numerical examples are given to illustrate the good performance of the method.

A predator-prey model was extended to include nonlinear harvesting of the predator guided by its population, such that harvesting is only implemented if the predator population exceeds an economic threshold. Theoretical results showed that the har-vesting system undergoes multiple bifurcations, including fold, supercritical Hopf, Bogdanov-Takens and cusp bifurcations. We determine stability and dynamical be-haviors of the equilibrium of this system. Numerical simulation results are given to support our theoretical results.

In this paper, a numerical method is developed and analyzed for solving a class of fractional optimal control problems (FOCPs) with vector state and control functions using polynomial approximation. The fractional derivative is considered in the Caputo sense. To implement the proposed numerical procedure, the Ritz spectral method with Bernoulli polynomials basis is applied. By applying the Bernoulli polynomials and using the numerical estimation of the unknown functions, the FOCP is reduced to solve a system of algebraic equations. By rigorous proofs, the convergence of the numerical method is derived for the given FOCP. Moreover, a new fractional operational matrix compatible with the proposed spectral method is formed to ease the complexity in the numerical computations. At last, several test problems are provided to show the applicability and e, ectiveness of the proposed scheme numerically

We consider a , rst-order partial di, erential equation with constant irreversible co-e, cients in a Banach space in the regular case. The equation is split into equations in subspaces, in which non-degenerate subsystems are obtained. We obtain an ana-lytical solution of each system with Showalter-type conditions. Finally, an example is given to illustrate the theoretical results.

The present paper aims to get through a class of fractional optimal control prob-lems (FOCPs). Furthermore, the fractional derivative portrayed in the Caputo sense through the dynamics of the system as fractional di, erential equation (FDE). Get-ting through the solution, , rstly the FOCP is transformed into a functional optimiza-tion problem. Then, by using known formulas for computing fractional derivatives of Legendre wavelets (LWs), this problem has been reduce to an equivalent system of algebraic equations. In the next step, we can simply solved this algebraic system. In the end, some examples are given to bring about the validity and applicability of this technique and the convergence accuracy.

In this paper, we study the quadratic rules for the numerical solution of Hammerstein integral equation based on spline quasi-interpolant. Also the convergence analysis of the methods are given. The theoretical behavior is tested on examples and it is shown that the numerical results con, rm theoretical part.

In this paper, we proposed an e, ective method based on the scaling function of Daubechies wavelets for the solution of the brachistochrone problem. An analytic technique for solving the integral of Daubechies scaling functions on dyadic intervals is investigated and these integrals are used to reduce the brachistochrone problem into algebraic equations. The error estimate for the brachistochrone problem is proposed and the numerical results are given to verify the e, ectiveness of our method.

The Black-Scholes equation is one of the most important mathematical models in option pricing theory, but this model is far from market realities and cannot show memory e, ect in the , nancial market. This paper investigates an American option based on a time-fractional Black-Scholes equation under the constant elasticity of variance (CEV) model, which parameters of interest rate and dividend yield sup-posed as deterministic functions of time, and the price change of the underlying asset follows a fractal transmission system. This model does not have a closed-form solution,hence, we numerically price the American option by using a compact di, er-ence scheme. Also, we compare the time-fractional Black-Scholes equation under the CEV model with its generalized Black-Scholes model as ,= 1 and ,= 0. Moreover, we demonstrate that the introduced di, erence scheme is unconditionally stable and convergent using Fourier analysis. The numerical examples illustrate the e, ciency and accuracy of the introduced di, erence scheme.

In this study, a collocation method based on Laguerre polynomials is presented to numerically solve systems of linear di, erential equations with variable coe, cients of high order. The method contains the following steps. Firstly, we write the Laguerre polynomials, their derivatives, and the solutions in matrix form. Secondly, the system of linear di, erential equations is reduced to a system of linear algebraic equations by means of matrix relations and collocation points. Then, the conditions in the problem are also written in the form of matrix of Laguerre polynomials. Hence, by using the obtained algebraic system and the matrix form of the conditions, a new system of linear algebraic equations is obtained. By solving the system of the obtained new algebraic equation, the coe, cients of the approximate solution of the problem are determined. For the problem, the residual error estimation technique is o, ered and approximate solutions are improved. Finally, the presented method and error estimation technique are demonstrated with the help of numerical examples. The results of the proposed method are compared with the results of other methods.

We present and investigate the delayed model of HIV infection for drug therapy. The stability of the equilibrium states, disease free and infected equilibrium states are derived and the existence of Hopf bifurcation analysis is studied. We show that the system is asymptotically stable and the stability is lost in a range due to length of the delay, then Hopf bifurcation occurs when ,exceeds the critical value. At last numerical simulations are provided to verify the theoretical results.

In this paper, we study the existence of a nontrival weak solution for a class of Kirch-ho,type problems with singular potentials and critical exponents. The proofs are essentially based on an appropriated truncated argument, Ca, arelli-Kohn-Nirenberg inequalities, combined with a variant of the concentration compactness principle. We also get a priori estimates of the obtained solution.

In this paper, we investigate stability in distribution of neutral stochastic functional di, erential equations with in, nite delay (NSFDEwID) at the state space Cr. We drive a su, cient strong monotone condition for the existence and uniqueness of the global solutions of NSFDEwID in the state space Cr. We also address the stability of the solution map xt and illustrate the theory with an example.

In this work, we consider a logarithmic nonlinear viscoelastic wave equation with a delay term in a bounded domain. We obtain the local existence of the solution by using the Faedo-Galerkin approximation. Then, under suitable conditions, we prove the blow up of solutions in , nite time.

Fractional integral operators play an important role in generalizations and extensions of various subjects of sciences and engineering. This research is the study of bounds of Riemann-Liouville fractional integrals via (h 􀀀,m)-convex functions. The author succeeded to , nd upper bounds of the sum of left and right fractional integrals for (h􀀀, m)-convex function as well as for functions which are deducible from aforementioned function (as comprise in Remark 1. 2). By using (h 􀀀,m)-convexity of jf′, j a modulus inequality is established for bounds of Riemann-Liouville fractional integrals. Moreover, a Hadamard type inequality is obtained by imposing an additional condition. Several special cases of the results of this research are identi, ed.