Implicit–explicit predictor–corrector schemes for nonlinear parabolic differential equations

In the present paper, a family of predictor–corrector (PC) schemes are developed for the numerical solution of nonlinear parabolic differential equations. Iterative processes are avoided by use of the implicit–explicit (IMEX) methods. Moreover, compared to the predictor schemes, the proposed methods usually have superior accuracy and stability properties. Some confirmation of these are illustrated by using the schemes on the well-known Fisher’s equation.     

Numerical solution of stochastic differential equations by second order Runge–Kutta methods

In this paper we propose the numerical solutions of stochastic initial value problems via random Runge–Kutta methods of the second order and mean square convergence of these methods is proved. A random mean value theorem is required and established. The concept of mean square modulus of continuity is also introduced. Expectation and variance of the approximating process are computed. Numerical examples show that the approximate solutions have a good degree of accuracy.     

Strong predictor–corrector Euler–Maruyama methods for stochastic differential equations with Markovian switching

In this paper numerical methods for solving stochastic differential equations with Markovian switching (SDEwMSs) are developed by pathwise approximation. The proposed family of strong predictor–corrector Euler–Maruyama methods is designed to overcome the propagation of errors during the simulation of an approximate path. This paper not only shows the strong convergence of the numerical solution to the exact solution but also reveals the order of the error under some conditions on the coefficient functions. A natural analogue of the p$p$-stability criterion is studied. Numerical examples are given to illustrate the computational efficiency of the new predictor–corrector Euler–Maruyama approximation.     

Numerical solution of singular boundary value problems using Green’s function and improved decomposition method

In this work, an effective technique for solving a class of singular two point boundary value problems is proposed. This technique is based on the Adomian decomposition method (ADM) and Green’s function. The technique relies on constructing Green’s function before establishing the recursive scheme for the solution components. In contrast to the existing recursive schemes based on ADM, the proposed recursive scheme avoids solving a sequence of nonlinear algebraic or transcendental equations for the undetermined coefficients. The approximate solution is obtained in the form of series with easily calculable components. For the completeness, the convergence and error analysis of the proposed scheme is supplemented. Moreover, the numerical examples are included to demonstrate the accuracy, applicability, and generality of the proposed scheme. The results reveal that the method is very effective, straightforward, and simple.     

Numerical stability of Grünwald–Letnikov method for time fractional delay differential equations

BIT Numerical Mathematics - This paper is concerned with the numerical stability of time fractional delay differential equations (F-DDEs) based on Grünwald–Letnikov (GL) approximation...     

Numerical solution of singular IVPs of Lane–Emden type using a modified Legendre-spectral method

In this paper, a numerical method for singular initial value problems of the Lane–Emden type in the second-order ordinary differential equations is proposed. The method changes solving the equation to solving a Volterra integral equation. We have applied the improved Legendre-spectral method to solve Lane–Emden type equations. The Legendre Gauss points are used as collocation nodes and Lagrange interpolation is employed in Volterra term. The results reveal that the method is effective, simple and accurate.     

Numerical simulation of blowup in nonlocal reaction–diffusion equations using a moving mesh method

In this paper we implement the moving mesh PDE method for simulating the blowup in reaction–diffusion equations with temporal and spacial nonlinear nonlocal terms. By a time-dependent transformation, the physical equation is written into a Lagrangian form with respect to the computational variables. The time-dependent transformation function satisfies a parabolic partial differential equation — usually called moving mesh PDE (MMPDE). The transformed physical equation and MMPDE are solved alternately by central finite difference method combined with a backward time-stepping scheme. The integration time steps are chosen to be adaptive to the blowup solution by employing a simple and efficient approach. The monitor function in MMPDEs plays a key role in the performance of the moving mesh PDE method. The dominance of equidistribution is utilized to select the monitor functions and a formal analysis is performed to check the principle. A variety of numerical examples show that the blowup profiles can be expressed correctly in the computational coordinates and the blowup rates are determined by the tests.     

Numerical solution of Volterra–Fredholm integral equations by moving least square method and Chebyshev polynomials

In this paper, we use a method based on moving least squares method and Chebyshev polynomials for numerical solution of Volterra–Fredholm integral equations of the second kind. The main advantage of this method is that it does not need a mesh, neither for purposes of interpolation nor for integration. The convergence of the method is investigated, and finally some examples are given to show the applicability of the method.     

A new predictor–corrector method for solving unconstrained minimization problems

This paper presents a new predictor–corrector method for finding a local minimum of a twice continuously differentiable function. The method successively constructs an approximation to the solution curve and determines a predictor on it using a technique similar to that used in trust region methods for unconstrained optimization. The proposed predictor is expected to be more effective than Euler's predictor in the sense that the former is usually much closer to the solution curve than the latter for the same step size. Results of numerical experiments are reported to demonstrate the effectiveness of the proposed method.     

Numerical solution of nonlinear Jaulent–Miodek and Whitham–Broer–Kaup equations

In this work we investigate the numerical solution of Jaulent–Miodek (JM) and Whitham–Broer–Kaup (WBK) equations. The proposed numerical schemes are based on the fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original partial differential equations (PDEs) with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. After transforming the equations to a system of ODEs, the linear operator in JM equation is diagonal but in WBK equation is not diagonal. However for WBK equation we can also implement the methods such as diagonal case which reduces the CPU time. Comparing numerical solutions with analytical solutions demonstrates that those methods are accurate and readily implemented.     

Numerical solution of the Yukawa-coupled Klein–Gordon–Schrödinger equations via a Chebyshev pseudospectral multidomain method

The Klein–Gordon–Schrödinger equations describe a classical model of the interaction between conservative complex neutron field and neutral meson Yukawa in quantum field theory. In this paper, we study the long-time behavior of solutions for the Klein–Gordon–Schrödinger equations. We propose the Chebyshev pseudospectral collocation method for the approximation in the spatial variable and the explicit Runge–Kutta method in time discretization. In comparison with the single domain, the domain decomposition methods have good spatial localization and generate a sparse space differentiation matrix with high accuracy. In this study, we choose an overlapping multidomain scheme. The obtained numerical results show the Pseudospectral multidomain method has excellent long-time numerical behavior and illustrate the effectiveness of the numerical scheme in controlling two particles. Some comparisons with single domain pseudospectral and finite difference methods will be also investigated to confirm the efficiency of the new procedure.     

Reduced differential transform method for nonlinear integral member of Kadomtsev–Petviashvili hierarchy differential equations

The main objective of this paper is to use the reduced differential to transform method (RDTM) for finding the analytical approximate solutions of two integral members of nonlinear Kadomtsev–Petviashvili (KP) hierarchy equations. Comparing the approximate solutions which obtained by RDTM with the exact solutions to show that the RDTM is quite accurate, reliable and can be applied for many other nonlinear partial differential equations. The RDTM produces a solution with few and easy computation. This method is a simple and efficient method for solving the nonlinear partial differential equations. The analysis shows that our analytical approximate solutions converge very rapidly to the exact solutions.     

Use of the D‐method for differential equations

Harmful effects are pointed out of the use of the D‐method for finding particular integrals of linear differential equations with constant coefficients. These particularly apply for students without much mathematical aptitude, and for those who will go on to study quantum mechanics. An alternative method is outlined.     

Numerical simulation of sphere water entry problem using Eulerian–Lagrangian method

This paper looks at the hydrodynamic’s numerical simulation of a free-falling sphere impacting the free surface of water by using the coupled Eulerian–Lagrangian (CEL) formulation included in the commercial software ABAQUS. A 3D model of a sphere with an unsteady viscous transient flow condition is used for numerical simulation. The simulation is performed for sphere with different density. The simulation results are verified by showing the computed shape of the air cavity, displacement of sphere, pinch-off time and depth that agree well with experimental results.     

Numerical method for the KdV–Kawahara and Benney–Lin equations

We are concerned with the initial-boundary-value problem associated to the Korteweg – de Vries – Kawahara (KdVK) equation and Benney – Lin (BL) equation, which are transport equations perturbed by dispersive terms of 3rd and 5th order and a term of 4th order in the case of (BL) equation. These equations appear in several fluid dynamics problems. We obtain local smoothing effects that are uniform with respect to the size of the interval. We also propose a simple finite-difference scheme for the problem and prove its stability. Finally, we give some numerical examples. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Error estimates of the backward Euler–Maruyama method for multi-valued stochastic differential equations

BIT Numerical Mathematics - In this paper we derive error estimates of the backward Euler–Maruyama method applied to multi-valued stochastic differential equations. An important example of...