Improved King’s methods with optimal order of convergence based on rational approximations

In this paper, we present three-point and four-point methods for solving nonlinear equations. The methodology is based on King’s family of fourth order methods [R.F. King, A family of fourth order methods for nonlinear equations, SIAM J. Numer. Anal. 10 (1973) 876–879] and further developed by using rational function approximations. The three-point method requires four function evaluations and has the order of convergence eight, whereas the four-point method requires five function evaluations and has the order of convergence sixteen. Therefore, the methods are optimal in the sense of Kung–Traub hypothesis. The proposed schemes are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified in the examples.     

Existence of solutions to nonlinear advection-diffusion equation applied to Burgers’ equation using Sinc methods

This paper has two objectives. First, we prove the existence of solutions to the general advection-diffusion equation subject to a reasonably smooth initial condition. We investigate the behavior of the solution of these problems for large values of time. Secondly, a numerical scheme using the Sinc-Galerkin method is developed to approximate the solution of a simple model of turbulence, which is a special case of the advection-diffusion equation, known as Burgers’ equation. The approximate solution is shown to converge to the exact solution at an exponential rate. A numerical example is given to illustrate the accuracy of the method.     

Solving Chisini’s functional equation

We investigate the n-variable real functions G that are solutions of the Chisini functional equation F(x) = F(G(x), . . . , G(x)), where F is a given function of n real variables. We provide necessary and sufficient conditions on F for the existence and uniqueness of solutions. When F is nondecreasing in each variable, we show in a constructive way that if a solution exists then a nondecreasing and idempotent solution always exists. We also provide necessary and sufficient conditions on F for the existence of continuous solutions and we show how to construct such a solution. We finally discuss a few applications of these results.     

On a nonhomogeneous Burgers’ equation

In this paper the Cauchy problem for the following nonhomogeneous Burgers’ equation is considered : (1)u _t +uu _x =μu _xx −kx,x ∈R,t > 0, where μ and k are positive constants. Since the nonhomogeneous term kx does not belong to any L^p(R) space, this type of equation is beyond usual Sobolev framework in some sense. By Hopf-Cole transformation, (1) takes the form (2)ϕ _t −ϕ _xx = −x ² ϕ. With the help of the Hermite polynomials and their properties, (1) and (2) are solved exactly. Moreover, the large time behavior of the solutions is also considered, similar to the discussion in Hopf’s paper. Especially, we observe that the nonhomogeneous Burgers’ equation (1) is nonlinearly unstable.     

Solving Schrödinger equation by using modified variational iteration and homotopy analysis methods

In this paper, a nonlinear Schr ö dinger equation is solved by using the variational iteration method (VIM), modified variational iteration method (MVIM) and homotopy analysis method (HAM) numerically. For each method, the approximate solution of this equation is calculated based on a recursive relation which its components are computed easily. The existence and uniqueness of the solution and the convergence of the proposed methods are proved. A numerical example is studied to demonstrate the accuracy of the given algorithms     

Normal form for the KdV–Burgers equation

The local dynamics of the KdV–Burgers equation with periodic boundary conditions is studied. A special nonlinear partial differential equation is derived that plays the role of a normal form, i.e., in the first approximation, it determines the behavior of all solutions of the original boundary value problem with initial conditions from a sufficiently small neighborhood of equilibrium.     

Taylor–Galerkin and Taylor-collocation methods for the numerical solutions of Burgers’ equation using B-splines

In this paper, for the numerical solution of Burgers’ equation, we give two B-spline finite element algorithms which involve a collocation method with cubic B-splines and a Galerkin method with quadratic B-splines. In time discretization of the equation, Taylor series expansion is used. In order to verify the stabilities of the purposed methods, von-Neumann stability analysis is employed. To see the accuracy of the methods, L₂ and L_∞ error norms are calculated and obtained results are compared with some earlier studies.     

Well-posedness of the stochastic KdV–Burgers equation

We are interested in rigorously proving the invariance of white noise under the flow of a stochastic KdV–Burgers equation. This paper establishes a result in this direction. After smoothing the additive noise (by a fractional spatial derivative), we establish (almost sure) local well-posedness of the stochastic KdV–Burgers equation with white noise as initial data. Next we observe that spatial white noise is invariant under the projection of this system to the first N>0$N>0$ modes of the trigonometric basis. Finally, we prove a global well-posedness result under an additional smoothing of the noise.     

Exact soliton solutions of the modified KdV–KP equation and the Burgers–KP equation by using the first integral method

In this paper, the first integral method is used to construct exact solutions of the modified KdV–KP equation and the Burgers–Kadomtsev–Petviashvili (Burgers–KP) equation. This method can be applied to nonintegrable equations as well as to integrable ones. This method is based on the theory of commutative algebra.     

Finite difference approximations for the fractional Fokker–Planck equation

The fractional Fokker–Planck equation has been used in many physical transport problems which take place under the influence of an external force field. In this paper we examine some practical numerical methods to solve a class of initial-boundary value problems for the fractional Fokker–Planck equation on a finite domain. The solvability, stability, consistency, and convergence of these methods are discussed. Their stability is proved by the energy method. Two numerical examples are also presented to evaluate these finite difference methods against the exact analytical solutions.     

Numerical solution of the coupled viscous Burgers’ equation

In the present paper, a numerical method is proposed for the numerical solution of a coupled system of viscous Burgers’ equation with appropriate initial and boundary conditions, by using the cubic B-spline collocation scheme on the uniform mesh points. The scheme is based on Crank–Nicolson formulation for time integration and cubic B-spline functions for space integration. The method is shown to be unconditionally stable using von-Neumann method. The accuracy of the proposed method is demonstrated by applying it on three test problems. Computed results are depicted graphically and are compared with those already available in the literature. The obtained numerical solutions indicate that the method is reliable and yields results compatible with the exact solutions.     

Multi-soliton solution,rational solution of the Boussinesq–Burgers equations

In this paper we consider the Boussinesq–Burgers equations and establish the transformation which turns the Boussinesq–Burgers equations into the single nonlinear partial differential equation, then we obtain an auto-Bäcklund transformation and abundant new exact solutions, including the multi-solitary wave solution and the rational series solutions. Besides the new trigonometric function periodic solutions are obtained by using the generalized tan h method.     

Neimark–Sacker bifurcation of a third-order rational difference equation

In this paper, we investigate the existence and direction of the Neimark–Sacker bifurcation of a third-order rational difference equation with positive parameters. Firstly, it is found that there exists a Neimark–Sacker bifurcation when the parameter passes a critical value by analysing the characteristic equation. Secondly, the explicit algorithm for determining the direction and stability of the Neimark–Sacker bifurcations is derived by using the normal form theory. Finally, computer simulations are performed to illustrate the analytical results found.     

Solving a Hamilton–Jacobi–Bellman equation with constraints

Optimal investment strategies for an insurer with state-dependent constraints are computed via a recursive finite difference solution to the corresponding discretized Hamilton–Jacobi–Belman equation. Convergence is derived from viscosity solution arguments. For this, a comparison result is given which is similar to the result given by Azcue and Muler [Ann. Appl. Probab. 20 (2010), pp. 1253–1302].     

Numerical approximations for the nonlinear time fractional reaction–diffusion equation

In this article, we propose an implicit pseudospectral scheme for nonlinear time fractional reaction–diffusion equations with Neumann boundary conditions, which is based upon Gauss–Lobatto–Legendre–Birkhoff pseudospectral method in space and finite difference method in time. A priori estimate of numerical solution is given firstly. Then the existence of numerical solution is proved by Brouwer fixed point theorem and the uniqueness is obtained. It is proved rigorously that the fully discrete scheme is unconditionally stable and convergent. Furthermore, we develop a modified scheme by adding correction terms for the problem with nonsmooth solutions. Numerical examples are given to verify the theoretical analysis.     

Interior estimates for the n-dimensional Abreuʼs equation

We study the Abreu?s equation in n-dimensional polytopes and derive interior estimates of solutions under the assumption of the uniform K-stability.     

Homogenization of Richardsʼ equation of van Genuchten–Mualem model

This paper is devoted to the homogenization of Richards? equation of van Genuchten–Mualem model, which is a nonlinear degenerate parabolic differential equation. It is usually used to model the motion of saturated–unsaturated water flow in porous media. We firstly apply the Kirchhoff transformation to the equation and obtain a simpler equivalent equation with a linear oscillated diffusion term. Then under the real assumption for van Genuchten–Mualem model, we obtain the homogenized equation based on the two-scale convergence theory. Some results on the first order corrector are also presented.     

N-cyclic functions and multiple subharmonic solutions of Duffingʼs equation

We introduce, in the abstract framework of finite isometry groups on a Hilbert space, a generalization of antiperiodicity called N-cyclicity. The non-existence of N-cyclic solutions of a certain type for the autonomous ODE $x^{″}+g(x)=0$ implies the existence of N different subharmonic solutions for some forced equations of the type $x^{″}+g(x)+c{x}^{\prime }=\epsilon f(t)$ where c and ε are some positive constants and f is, for instance, a sinusoidal function.