Computational Methods for Differential Equations
Year:2019 | Volume:7 | Issue:3|Papers Count:12

In this paper, we are going to obtain the soliton solution of the generalized Rosenau-Kawahara-RLW equation that describes the dynamics of shallow water waves in oceans and rivers. We confirm that our new algorithm is energy-preserved and unconditionally stable. In order to determine the performance of our numerical algorithm, we have computed the error norms L2 and L∞ . Convergence of full discrete scheme is firstly studied. Numerical experiments are implemented to validate the energy conservation and effectiveness for longtime simulation. The obtained numerical results have been compared with a study in the literature for similar parameters. This comparison clearly shows that our results are much better than the other results.

In this paper, an implicit finite difference scheme is proposed for the numerical solution of stochastic partial differential equations (SPDEs) of Itˆ o type. The consistency, stability, and convergence of the scheme are analyzed. Numerical experiments are included to show the efficiency of the scheme.

During the past years, a wide range of distinct approaches has been exerted to solve the nonlinear fractional differential equations (NLFDEs). In this paper, the invariant subspace method (ISM) in conjunction with the fractional Sumudu’ s transform (FST) in the conformable context is formally adopted to deal with a nonlinear conformable time-fractional dispersive equation of the fifth-order. As an outcome, a new exact solution of the model is procured, corroborating the exceptional performance of the hybrid scheme.

In this paper, we consider a mathematical model of leptospirosis disease which is an infectious disease. The model we are considering is a system of nonlinear ordinary differential equations and it is difficult to find exact solution. He’ s homotopy perturbation method is employed to compute an approximation to the solution of the system of nonlinear ordinary differential equations governing on the problem. The findings obtained by HPM are compared with nonstandard finite difference (NSFD) and Runge-Kutta fourth order (RK4) methods. Some plots are presented to show the reliability and simplicity of the method.

In this work, a general mathematical model of Diffusion Tensor Magnetic Resonance Imaging is formulated as an inverse problem. An effective numerical approach based on space marching method and mollification scheme is established to solve this problem. Convergence and stability of proposed approach are established. Using two test problems, the robustness and ability of the numerical approach is investigated.

In this study, we introduce a novel hybrid method based on a simple graph along with operational matrix to solve the combined functional neutral differential equations with several delays. The matrix relations of the matching polynomials of complete and path graphs are employed in the matrix-collocation method. We improve the obtained solutions via an error analysis technique. The oscillation of them on time interval is also estimated by coupling the method with Laplace-Pade technique. We develop a general computer program and so we can efficiently monitor the precision of the method. We investigate a convergence rate of the method by constructing a formula based on the residual function. Eventually, an algorithm is described to show the easiness of the method.

This paper is concerned with existence of three solutions for non-local fourth-order Kirchhoff systems with Navier boundary conditions. Our technical approach is based on variational methods and the theory of the variable exponent Sobolev spaces.

In this paper, we developed a collocation method based on cubic B-spline for solving nonlinear inverse parabolic partial differential equations as the following form ut = [f(u) ux]x + φ (x, t, u, ux), 0 < x < 1, 0  t  T, where f(u) and φ are smooth functions defined on R. First, we obtained a time discrete scheme by approximating the first-order time derivative via forward finite difference formula, then we used cubic B-spline collocation method to approximate the spatial derivatives and Tikhonov regularization method for solving produced illposed system. It is proved that the proposed method has the order of convergence O(k+h2). The accuracy of the proposed method is demonstrated by applying it on three test problems. Figures and comparisons have been presented for clarity. The aim of this paper is to show that the collocation method based on cubic B-spline is also suitable for the treatment of the nonlinear inverse parabolic partial differential equations.

In this paper, we investigate the dynamical complexities of a prey predator model prey refuge providing additional food to predator. We determine dynamical behaviors of the equilibria of this system and characterize codimension 1 and codimension 2 bifurcations of the system analytically. Hopf bifurcation conditions are derived analytically. We especially approximate a family of limit cycles emanating from a Hopf point. The analytical results are in well agreement with the numerical simulation results. Our bifurcation analysis indicates that the system exhibits numerous types of bifurcation phenomena, including Hopf, and Bogdanov-Takens bifurcations.

In this paper, we use the three critical points theorem attributed to B. Ricceri in order to establish existence of three distinct solutions for the following boundary value problem: 8<:
pu = a(x)jujp􀀀 2u in , jrujp􀀀 2ru:  =  f(x; u) on @.

In this paper, a modification of the Legendre collocation method is used for solving the space fractional differential equations. The fractional derivative is considered in the Caputo sense along with the finite difference and Legendre collocation schemes. The numerical results obtained by this method have been compared with other methods. The results show the capability and efficiency of the proposed method.

In this paper, we apply a numerical scheme for the solution of a second order partial integro-differential equation with a weakly singular kernel. In the time direction, the backward Euler method time-stepping is used to approximate the differential term and the cubic B-splines is applied to the space discretization. Detailed discrete schemes, the convergence and the stability of the method is demonstrated. Next, the computational efficiency and accuracy of the method are examined by the numerical results.