Regularity property of solution to two-parameter stochastic volterra equation with non-lipschitz coefficients

This article proves the existence and uniqueness of solution to two-parameter stochastic Volterra equation with non-Lipschitz coefficients and driven by Brownian sheet, where the main tool is Bihari's inequality in the plane. Moreover, we also discuss the time regularity property of the solution by Kolmogorov's continuity criterion.     

A numerical algorithm for nonlinear parabolic equations with highly oscillating coefficients

A numerical algorithm is constructed for the solution to a class of nonlinear parabolic operators in the case of homogenization. We consider parabolic operators of the form d/dt + A_ϵ, where A_ϵ is monotone. More precisely, we consider the case when A_ϵu=−div (a(x/ϵ, e/ϵ^k) |Du|^p−2Du), where p≥2 and k>0. © 1996 John Wiley & Sons, Inc.     

A new method to solve the second-order linear difference equations with constant coefficients

In this paper, we give a new and straightforward method to solve the non-homogeneous second-order linear difference equations with constant coefficients. It is new because it does not require the uniqueness theorem of the solution of the problem of initial values. Neither does it require a fundamental system of solutions, nor the method of variation of parameters. Moreover, we get a unique formula that expresses the general solution independently of the multiplicities of the roots of the characteristic equation.     

Convergence of a numerical scheme for SPDEs with correlated noise and global Lipschitz coefficients

The aim of this paper is to investigate the pathwise numerical solution of semilinear parabolic stochastic partial differential equations (SPDEs) with colored noise instead of the usual space–time white noise. We estimate the numerical solution in the L^∞ topology by a method that takes advantages of the smoothing effect of the dominant linear operator. We consider the case the covariance operator of the forcing does not necessarily commute with the linear operator of the SPDE because of the fact that the Brownian motions are not necessarily independent. We show convergence of this method, and numerical examples give insight into the reliability of the theoretical study. Copyright © 2015 John Wiley & Sons, Ltd.     

A multi-iterate method to solve systems of nonlinear equations

We propose an extension of secant methods for nonlinear equations using a population of previous iterates. Contrarily to classical secant methods, where exact interpolation is used, we prefer a least squares approach to calibrate the linear model. We propose an explicit control of the numerical stability of the method.     

A new method to solve the damped nonlinear Klein-Gordon equation

This paper discusses a damped nonlinear Klein-Gordon equation in the reproducing kernel space and provides a new method for solving the damped nonlinear Klein-Gordon equation based on the reproducing kernel space.Two numerical examples are given for illustrating the feasibility and accuracy of the method.     

A numerical scheme for periodic travelling-wave simulations in some nonlinear dispersive wave models

A numerical method for simulating periodic travelling-wave solutions of some nonlinear dispersive wave equations is proposed. The construction of the scheme is based on an efficient computation of the elements that characterize these solutions: the initial profile and the velocity of the wave.     

A numerical scheme based on a solution of nonlinear advection–diffusion equations

A mathematical formulation of the two-dimensional Cole–Hopf transformation is investigated in detail. By making use of the Cole–Hopf transformation, a nonlinear two-dimensional unsteady advection–diffusion equation is transformed into a linear equation, and the transformed equation is solved by the spectral method previously proposed by one of the authors. Thus a solution to initial value problems of nonlinear two-dimensional unsteady advection–diffusion equations is derived. On the base of the solution, a numerical scheme explicit with respect to time is presented for nonlinear advection–diffusion equations. Numerical experiments show that the present scheme possesses the total variation diminishing properties and gives solutions with good quality.     

Some remarks on stochastic differential equations in the plane with local lipschitz coefficients

We prove the existence and uniqueness of a local solution for stochastic differential equations in the plane with local Lipschitz coefficients, and state the existence of random points where the solution explodes. A sufficient condition to obtain a global solution is given.     

A numerical approach to solve the model of an electromechanical system

In this paper, the model of an electromechanical system, which is a system of linear differential equations, is studied. Haar wavelet collocation method (HWCM) is applied for finding the approximate solution of the model. HWCM reduces the system of the model into a matrix‐vector form that contains the unknown Haar coefficients, and these coefficients are easily calculated. To demonstrate the validity and applicability of HWCM, numerical solutions of the system for different parameter values in the system are presented. The obtained results demonstrate the efficiency and accuracy of the method. All of the computations are performed via a program written in Mathematica.     

The dirichlet problem in lipschitz domains for higher order elliptic systems with rough coefficients

We study the Dirichlet problem, in Lipschitz domains and with boundary data in Besov spaces, for divergence form strongly elliptic systems of arbitrary order with bounded, complex-valued coefficients. A sharp corollary of our main solvability result is that the operator of this problem performs an isomorphism between weighted Sobolev spaces when its coefficients and the unit normal of the boundary belong to the space VMO.     

ACCPM with a nonlinear constraint and an active set strategy to solve nonlinear multicommodity flow problems

This paper proposes an implementation of a constrained analytic center cutting plane method to solve nonlinear multicommodity flow problems. The new approach exploits the property that the objective of the Lagrangian dual problem has a smooth component with second order derivatives readily available in closed form. The cutting planes issued from the nonsmooth component and the epigraph set of the smooth component form a localization set that is endowed with a self-concordant augmented barrier. Our implementation uses an approximate analytic center associated with that barrier to query the oracle of the nonsmooth component. The paper also proposes an approximation scheme for the original objective. An active set strategy can be applied to the transformed problem: it reduces the dimension of the dual space and accelerates computations. The new approach solves huge instances with high accuracy. The method is compared to alternative approaches proposed in the literature. An erratum to this article can be found at     

Quintic trigonometric spline based numerical scheme for nonlinear modified Burgers' equation

This paper presents a numerical method based on quintic trigonometric B‐splines for solving modified Burgers' equation (MBE). Here, the MBE is first discretized in time by Crank–Nicolson scheme and the resulting scheme is solved by quintic trigonometric B‐splines. The proposed method tackles nonlinearity by using a linearization process known as quasilinearization. A rigorous analysis of the stability and convergence of the proposed method are carried out, which proves that the method is unconditionally stable and has order of convergence O(h⁴ + k²). Numerical results presented are very much in accordance with the exact solution, which is established by the negligible values of L₂ and L_∞ errors. Computational efficiency of the scheme is proved by small values of CPU time. The method furnishes results better than those obtained by using most of the existing methods for solving MBE.     

A comparison of the Newton-Krylov method with high order Newton-like methods to solve nonlinear systems

We compare the CPU time and error estimates of some variants of Newton method of the third and fourth-order convergence with those of the Newton-Krylov method used to solve systems of nonlinear equations. By expanding some numerical experiments we show that the use of Newton-Krylov method is better in the cost and accuracy points of view than the use of other high order Newton-like methods when the system is sparse and its size is large.     

A weak condition for secant method to solve systems of nonlinear equations

In this paper, a new weak condition for the convergence of secant method to solve the systems of nonlinear equations is proposed. A convergence ball with the center x0 is replaced by that with xl, the first approximation generated by the secant method with the initial data x-1 and x0. Under the bounded conditions of the divided difference, a convergence theorem is obtained and two examples to illustrate the weakness of convergence conditions are provided. Moreover, the secant method is applied to a system of nonlinear equations to demonstrate the viability and effectiveness of the results in the paper.     

A numerical solution of singular integrodifferential equations with variable coefficients

A method for the numerical solution of singular integrodifferential equations is presented where the integrals are discretized by using a convenient quadrature rule. Then the problem is reduced to a system of linear algebraic equations by applying the discretized functional equation to appropriately selected collocation points. This technique constitutes an extension of an analogous method convenient for solving singular integral equations which was proposed by the authors.     

A numerical scheme for a class of nonlinear Fredholm integral equations of the second kind

In this paper an iterative approach for obtaining approximate solutions for a class of nonlinear Fredholm integral equations of the second kind is proposed. The approach contains two steps: at the first one, we define a discretized form of the integral equation and prove that by considering some conditions on the kernel of the integral equation, solution of the discretized form converges to the exact solution of the problem. Following that, in the next step, solution of the discretized form is approximated by an iterative approach. We finally on some examples show the efficiency of the proposed approach.