فهرست مطالب
Computational Methods for Differential Equations
Volume:11 Issue: 2, Spring 2023
 تاریخ انتشار: 1402/02/05
 تعداد عناوین: 15


Pages 207224The present study aims to investigate the optimal fractional order PID controller performance in the chaotic system of HIV disease fractional order using the Particle Swarm optimization and Genetic algorithm method. Differential equations were used to represent the chaotic behavior associated with HIV. The optimal fractional order of the PID controller was constructed, and its performance in the chaotic system with HIV fractional order was tested. Optimization methods were used to get PID control coefficients from particle swarm and genetic algorithms. Findings revealed that the equations for the HIV disease model are such that the system’s behavior is greatly influenced by the number of viruses produced by infected cells, such that if the number of viruses generated by infected cells exceeds 202, the disease’s behavior is such that the virus and disease spread. For varying concentrations of viruses, the controller created for this disease does not transmit the disease.Keywords: Chaotic System, Optimal Fractional Order Controller, genetic algorithm, Particle swarm optimization algorithm, HIV

Pages 225240In this paper, a new algorithm based on nonpolynomial spline is developed for the solution of higher order boundary value problems(BVPs). Employment of the method is done by decomposing the higher order BVP into a system of third order BVPs. Convergence analysis of the developed method is also discussed. The method is tested on higher order linear as well as nonlinear BVPs which shows the accuracy and efficiency of the method and also compared our results with some existing fourth order methods.Keywords: Nonpolynomial spline, Higherorder, Nonlinear, Convergence analysis, Boundary value problems

Pages 241253
This paper presents an efficient numerical method to solve two versions of the Duffing equation by the hybrid functions based on the combination of Blockpulse functions and Legendre polynomials. This method reduces the solution of the considered problem to the solution of a system of algebraic equations. Moreover, the convergence of the method is studied. Some examples are given to demonstrate the applicability and effectiveness of the proposed method. Also, the obtained results are compared with some other results.
Keywords: Integral boundary conditions, Boundary value problem, Duffing equation, Hybrid functions, Legendre polynomials 
Pages 254262In this paper, we study the bifurcation of nontrivial steady state solutions for a crossdiffusion preypredator model with homogeneous Neumann boundary conditions. The existence of positive steady state solutions near a bifurcation point is proved using a crossing curve bifurcation theorem. We consider a situation where the transversality condition is not satisfied. Unlike the case in saddlenode bifurcation, the solution set is a pair of transversally intersecting curves.Keywords: Steady state bifurcation, Cross diffusion, Preypredator model

Pages 263280In this study, the unified and improved Fexpansion methods are applied to derive exact traveling wave solutions of the simplified modified CamassaHolm (SMCH) equation. The current methods can calculate all branches of solutions at the same time, even if several solutions are quite near and therefore impossible to identify via numerical methods. All obtained solutions are given by hyperbolic, trigonometric, and rational function solutions which obtained solutions are useful for reallife problems in fluid dynamics, optical fibers, plasma physics and so on. The twodimensional (2D) and threedimensional (3D) graphs of the obtained solutions are plotted. Finally, we can state that these strategies are extremely successful, dependable, and simple. These ideas might potentially be applied to many nonlinear evolution models in mathematics and physics.Keywords: SMCH equation, the improved Fexpansion method, the unified method, Symbolic computation, Exact solution, solitary wave

Pages 281290
A numerical method based on the Haar wavelet is introduced in this study for solving the partial differential equation which arises in the pricing of European options. In the first place, and due to the change of variables, the related partial differential equation (PDE) converts into a forward time problem with a spatial domain ranging from 0 to 1. In the following, the Haar wavelet basis is used to approximate the highest derivative order in the equation concerning the spatial variable. Then the lower derivative orders are approximated using the Haar wavelet basis. Finally, by substituting the obtained approximations in the main PDE and doing some computations using the finite differences approach, the problem reduces to a system of linear equations that can be solved to get an approximate solution. The provided examples demonstrate the effectiveness and precision of the method.
Keywords: European option, Haar wavelet, Option pricing 
Pages 291302Exponential functions play an essential role in describing the qualitative properties of solutions of nabla fractional difference equations. In this article, we illustrate their asymptotic behavior. We know that these functions involve infinite series of ratios of gamma functions, and it is challenging to compute them. For this purpose, we propose a novel matrix technique to compute the addressed functions numerically. The results are supported by illustrative examples. The proposed method can be extended to obtain numerical solutions for nonhomogeneous nabla fractional difference equations.Keywords: Nabla fractional difference, Exponential function, triangular strip matrix, general solution, Asymptotic behaviour

Pages 303318In this paper, we use the rational radial basis functions ( RRBFs) method to solve the Kortewegde Vries (KdV) equation, particularly when the equation has a solution with steep front or sharp gradients. We approximate the spatial derivatives by RRBFs method then we apply an explicit fourthorder RungeKutta method to advance the resulting semidiscrete system in time. Numerical examples show that the presented scheme preserves the conservation laws and the results obtained from this method are in good agreement with analytical solutions.Keywords: KdV equation, RBF, rational radial basis function method, RungeKutta method

Pages 319331This work deals with the existence of periodic solutions (EPSs) to a third order nonlinear delay differential equation (DDE) with multiple constant delays. For the considered DDE, a theorem is proved, which includes sufficient criteria related to the EPSs. The technique of the proof depends on LyapunovKrasovskiˇı functional (LKF) approach. The obtained result extends and improves some results that can be found in the literature. In a particular case of the considered DDE, an example is provided to show the applicability of the main result of this paper.Keywords: existence, periodic solutions, Differential equation, third order, delay, LKF

Pages 332344In this paper, we prove the existence and uniqueness of the solutions for a noninteger high order boundary value problem which is subject to the Caputo fractional derivative. The boundary condition is a nonlocal type. Analytically, we introduce the fractional Green’s function. The main principle applied to simulate our results is the Banach contraction fixed point theorem. We deduce this paper by presenting some illustrative examples. Furthermore, it is presented a numerical based semianalytical technique to approximate the unique solution to the desired order of precision.Keywords: High order fractional differential equation, Caputo fractional derivative, Boundary value problem, Existence, uniqueness, fixed point theorem

Pages 345356This paper investigates the semianalytical solutions of linear and nonlinear Time Fractional KleinGordon equations with appropriate initial conditions to apply the New SumuduAdomian Decomposition method (NSADM). This paper shows the semianalytical as well as a graphical interpretation of the solution by using mathematical software “Mathematica Wolform” and considering Caputo’s sense derivatives to semianalytical results obtained by NSADM.Keywords: KleinGordon equation of fractional order, Adomian Decomposition Method, Sumudu transform, Caputoderivative

Pages 357368In this paper, a pseudospectral method is proposed for solving the nonlinear timefractional KleinGordon and sineGordon equations. The method is based on the Sinc operational matrices. A finite difference scheme is used to discretize the Caputo timefractional derivative, while the spatial derivatives are approximated by the Sinc method. The convergence of the full discretization of the problem is studied. Some numerical examples are presented to confirm the accuracy and efficiency of the proposed method. The numerical results are compared with the analytical solution and the reported results in the literature.Keywords: fractional differential equation, Nonlinear KleinGordon, sineGordon equations, Sinc operational matrices, Pseudospectral method, Convergence

Pages 369386In this article, we discuss the numerical solution of the nonlinear SineGordon equation in one and two dimensions and its coupled form. A differential quadrature technique based on a modified set of cubic Bsplines has been used. The chosen modification possesses the optimal accuracy order four in the spatial domain. The spatial derivatives are approximated by the differential quadrature technique, where the weight coefficients are calculated using this set of modified cubic Bsplines. This approximation will lead to the discretization of the problem in the spatial domain that gives a system of firstorder ordinary differential equations. This system is then solved using the SSPRK54 scheme to progress the solution to the next time level. The convergence of this numerical scheme solely depends on the differential quadrature and is found to give a stable solution. The order of convergence is calculated and is observed to be four. The entire computation is performed up to a large time level with an efficient speed. It is found that the computed solution is in good agreement with the exact one and the error comparison with similar works in the literature indicates the scheme outperforms.Keywords: SineGordon equation, Differential quadrature method, Modified cubic Bspline, SSPRK54

Pages 387398This paper deals with the generalization of the fractional operational matrix of Jacobi wavelets. The fractional population growth model was solved by using this operational matrix and compared with other existing methods to illustrate the applicability of the method. Then, convergence and error analysis of this procedure were studied.Keywords: Hybrid functions, Jacobi polynomials, Operational matrix, Population growth model, Wavelets

Pages 399411In this paper, we use a geometric approach based on the concepts of variational principle and moving frames to obtain the conservation laws related to the onedimensional nonlinear KleinGordon equation. Noether’s First Theorem guarantees conservation laws, provided that the Lagrangian is invariant under a Lie group action. So, for calculating conservation laws of the KleinGordon equation, we first present a Lagrangian whose EulerLagrange equation is the KleinGordon equation, and then according to Gon¸calves and Mansfield’s method, we obtain the space of conservation laws in terms of vectors of invariants and the adjoint representation of a moving frame, for that Lagrangian, which is invariant under a hyperbolic group action.Keywords: Nonlinear KleinGordon equation, Conservation laws, Moving frame, Differential invariants, Syzygy