Singular differential game numerical techniques

This paper develops a numerical technique to solve a class of zero-sum differential games with singular control. By using this technique and the application of inverse systems, a near-optimal closed-loop technique is developed to generate a numerical solution to this class of problems.     

On the continuous extension of Adams–Bashforth methods and the event location in discontinuous ODEs

The interpolation polynomial in the $k$-step Adams–Bashforth method may be used to compute the numerical solution at off grid points. We show that such a numerical solution is equivalent to the one obtained by the Nordsieck technique for changing the step size. We also provide an application of this technique to the event location in discontinuous differential systems.     

The exact solution of a class of Volterra integral equation with weakly singular kernel

In this paper, the weakly singular Volterra integral equations with an infinite set of solutions are investigated. Among the set of solutions only one particular solution is smooth and all others are singular at the origin. The numerical solutions of this class of equations have been a difficult topic to analyze and have received much previous investigation. The aim of this paper is to present a numerical technique for giving the approximate solution to the only smooth solution based on reproducing kernel theory. Applying weighted integral, we provide a new definition for reproducing kernel space and obtain reproducing kernel function. Using the good properties of reproducing kernel function, the only smooth solution is exactly expressed in the form of series. The n-term approximate solution is obtained by truncating the series. Meanwhile, we prove that the derivative of approximation converges to the derivative of exact solution uniformly. The final numerical examples compared with other methods show that the method is efficient.     

A numerical integrator for a model with a discontinuous sink term: the dynamics of the sexual phase of monogonont rotifera

We present an explicit numerical method especially adapted to approximate the solution of a nonlinear model with a discontinuous sink term: the dynamics of the sexual phase of monogonont rotifera. We show the effectiveness of this numerical technique in the simulation of the dynamics of the solutions. In particular, we show that this method provides a good approximation to the equilibrium solution of the problem. On the other hand, the numerical simulation presented validates the asymptotic behaviour of the numerical solution with regard to the theoretical one.     

Spectral-finite element method for solving two-dimensional vorticity equations

In this paper, we construct a spectral-finite element scheme for solving semi-periodical two-dimensional vorticity equations. The error between the genuine solution and approximate solution is estimated strictly. The numerical results show the advantages of such a method. The technique used in this paper can be easily generalized to three-dimensional problems.     

Numerical solution of Fokker‐Planck equation using the cubic B‐spline scaling functions

In this article a numerical technique is presented for the solution of Fokker‐Planck equation. This method uses the cubic B‐spline scaling functions. The method consists of expanding the required approximate solution as the elements of cubic B‐spline scaling function. Using the operational matrix of derivative, the problem will be reduced to a set of algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Polynomial scaling functions for numerical solution of generalized Kuramoto–Sivashinsky equation

The current paper proposes a technique for the numerical solution of generalized Kuramoto–Sivashinsky equation. The method is based on finite difference formula combined with the collocation method, which uses the polynomial scaling functions (PSF). Mentioned functions and their properties are employed to derive a general procedure for forming the operational matrix of PSFs. Using the operational matrix of derivative, we reduce the problem to a set of algebraic linear equations. An estimation of error bound for this method is presented. Some numerical example is included to demonstrate the validity and applicability of the technique. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by previous works and also it is efficient to use.     

Numerical solution of the Bagley–Torvik equation by the Bessel collocation method

In this article, a numerical technique is presented for the approximate solution of the Bagley–Torvik equation, which is a class of fractional differential equations. The basic idea of this method is to obtain the approximate solution in a generalized form of the Bessel functions of the first kind. For this purpose, by using the collocation points, the matrix operations and a generalization of the Bessel functions of the first kind, this technique transforms the Bagley–Torvik equation into a system of the linear algebraic equations. Hence, by solving this system, the unknown Bessel coefficients are computed. The reliability and efficiency of the proposed scheme are demonstrated by some numerical examples. Copyright © 2012 John Wiley & Sons, Ltd.     

A new approach for solving a class of nonlinear integro-differential equations

In this paper, a new technique for solving a class of quadratic integral and integro-differential equations is introduced. The main advantage of this technique is that it can replace the nonlinear problem by an equivalent linear one or by another simpler nonlinear one. The convergence of the series solution is proved. Convergence analysis is reliable enough to estimate the maximum absolute truncated error of the series solution. Some numerical examples are introduced to verify the efficiency of the new technique.     

Spectral corrections for Sturm–Liouville problems

The numerical solution of the Sturm–Liouville problem can be achieved using shooting to obtain an eigenvalue approximation as a solution of a suitable nonlinear equation and then computing the corresponding eigenfunction. In this paper we use the shooting method both for eigenvalues and eigenfunctions. In integrating the corresponding initial value problems we resort to the boundary value method. The technique proposed seems to be well suited to supplying a general formula for the global discretization error of the eigenfunctions depending on the discretization errors arising from the numerical integration of the initial value problems. A technique to estimate the eigenvalue errors is also suggested, and seems to be particularly effective for the higher-index eigenvalues. Numerical experiments on some classical Sturm–Liouville problems are presented.     

Numerical verification of solutions for variational inequalities

In this paper, we consider a numerical technique that enables us to verify the existence of solutions for variational inequalities. This technique is based on the infinite dimensional fixed point theorems and explicit error estimates for finite element approximations. Using the finite element approximations and explicit a priori error estimates for obstacle problems, we present an effective verification procedure that through numerical computation generates a set which includes the exact solution. Further, a numerical example for an obstacle problem is presented. Received October 28,1996 / Revised version received December 29,1997     

The equimaterial approach for the numerical solution of the one dimensional transient diffusion equation with zero flux boundary conditions

A new approach is proposed for the grid motion for the numerical solution of a general transient diffusion equation in one spatial dimension with zero flux boundary conditions. The new criterion for grid motion is that the solute amount contained in each discretization section should be a pre-described fraction of the total solute amount at each time step. This requirement is not explicitly enforced to the solution technique but it is implicitly included in the equation through the appropriate variable transformation. The results showed that although the technique leads to the required grid motion the numerical results are of pure quality due to the appearance of singularities during the variable transformation procedure. Nevertheless, it is shown that by appropriate numerical handling of the solution at the singularity region the technique can lead to accurate results and potentially can replace the existing moving grid algorithms at least for the particular problem at hand.     

On the Solution of Discontinuous IVPs by Adaptive Runge–Kutta Codes

In this paper an automatic technique for handling discontinuous IVPs when they are solved by means of adaptive Runge–Kutta codes is proposed. This technique detects, accurately locates and passes the discontinuities in the solution of IVPs by using the information generated by the code along the numerical integration together with a continuous interpolant of the discrete solution. A remarkable feature is that it does not require additional information on the location of the discontinuities. Some numerical experiments are presented to illustrate the reliability and efficiency of the proposed algorithms.     

Numerical studies of time-independent and time-dependent scattering by several elliptical cylinders

A numerical solution to the problem of time-dependent scattering by an array of elliptical cylinders with parallel axes is presented. The solution is an exact one, based on the separation-of-variables technique in the elliptical coordinate system, the addition theorem for Mathieu functions, and numerical integration. Time-independent solutions are described by a system of linear equations of infinite order which are truncated for numerical computations. Time-dependent solutions are obtained by numerical integration involving a large number of these solutions. First results of a software package generating these solutions are presented: wave propagation around three impenetrable elliptical scatterers. As far as we know, this method described has never been used for time-dependent multiple scattering.     

The comparison of two reliable methods for accurate solution of time‐fractional Kaup–Kupershmidt equation arising in capillary gravity waves

In this paper, we compared two different methods, one numerical technique, viz Legendre multiwavelet method, and the other analytical technique, viz optimal homotopy asymptotic method (OHAM), for solving fractional‐order Kaup–Kupershmidt (KK) equation. Two‐dimensional Legendre multiwavelet expansion together with operational matrices of fractional integration and derivative of wavelet functions is used to compute the numerical solution of nonlinear time‐fractional KK equation. The approximate solutions of time fractional Kaup–Kupershmidt equation thus obtained by Legendre multiwavelet method are compared with the exact solutions as well as with OHAM. The present numerical scheme is quite simple, effective, and expedient for obtaining numerical solution of fractional KK equation in comparison to analytical approach of OHAM. Copyright © 2016 John Wiley & Sons, Ltd.     

High‐resolution monotone schemes based on quasi‐characteristics technique

In this article, we consider a new technique that allows us to overcome the well‐known restriction of Godunov's theorem. According to Godunov's theorem, a second‐order explicit monotone scheme does not exist. The techniques in the construction of high‐resolution schemes with monotone properties near the discontinuities of the solution lie in choosing of one of two high‐resolution numerical solutions computed on different stencils. The criterion for choosing the final solution is proposed. Results of numerical tests that compare with the exact solution and with the numerical solution obtained by the first‐order monotone scheme are presented. © 2001 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 17: 262–276, 2001     

Simple layer potential method for domains having external corners

This note investigates the simple layer potential method for domains having external corners. The singular behaviour of simple layer density at the corners is studied for the Neumann problem of Helmholtz's equation. A numerical technique of solving the integral equation for this problem is proposed. This technique takes the singularity of the solution into consideration. Some numerical examples are given to show the applicability of the present method.     

Finite difference method combined with differential quadrature method for numerical computation of the modified equal width wave equation

The aim of this study is to improve the numerical solution of the modified equal width wave equation. For this purpose, finite difference method combined with differential quadrature method with Rubin and Graves linearizing technique has been used. Modified cubic B‐spline base functions are used as base function. By the combination of two numerical methods and effective linearizing technique high accurate numerical algorithm is obtained. Three main test problems are solved for various values of the coefficients. To observe the performance of the present method, the error norms of the single soliton problem which has analytical solution are calculated. Besides these error norms, three invariants are reported. Comparison of the results displays that our algorithm produces superior results than those given in the literature.     

Implicit a posteriori error estimates for the Maxwell equations

An implicit a posteriori error estimation technique is presented and analyzed for the numerical solution of the time-harmonic Maxwell equations using Nédélec edge elements. For this purpose we define a weak formulation for the error on each element and provide an efficient and accurate numerical solution technique to solve the error equations locally. We investigate the well-posedness of the error equations and also consider the related eigenvalue problem for cubic elements. Numerical results for both smooth and non-smooth problems, including a problem with reentrant corners, show that an accurate prediction is obtained for the local error, and in particular the error distribution, which provides essential information to control an adaptation process. The error estimation technique is also compared with existing methods and provides significantly sharper estimates for a number of reported test cases.