Numerical study of Fisher's reaction–diffusion equation by the Sinc collocation method

Fisher's equation, which describes a balance between linear diffusion and nonlinear reaction or multiplication, is studied numerically by the Sinc collocation method. The derivatives and integrals are replaced by the necessary matrices, and a system of algebraic equations is obtained to approximate solution of the problem. The error in the approximation of the solution is shown to converge at an exponential rate. Numerical examples are given to illustrate the accuracy and the implementation of the method, the results show that any local initial disturbance can propagate with a constant limiting speed when time becomes sufficiently large. Both the limiting wave fronts and the limiting speed are independent of the initial values.     

Quintic B-spline collocation method for numerical solution of the Kuramoto–Sivashinsky equation

In this paper, the quintic B-spline collocation scheme is implemented to find numerical solution of the Kuramoto–Sivashinsky equation. The scheme is based on the Crank–Nicolson formulation for time integration and quintic B-spline functions for space integration. The accuracy of the proposed method is demonstrated by four test problems. The numerical results are found to be in good agreement with the exact solutions. Results are also shown graphically and are compared with results given in the literature.     

Numerical conservative solutions of the Hunter–Saxton equation

BIT Numerical Mathematics - In the article a convergent numerical method for conservative solutions of the Hunter–Saxton equation is derived. The method is based on piecewise linear...     

Numerical simulation for the solitary wave of Zakharov–Kuznetsov equation based on lattice Boltzmann method

The Zakharov–Kuznetsov equation is considered, which is an equation describing two dimensional weakly nonlinear ion-acoustic waves in plasma. We focus on using the lattice Boltzmann method to study the Zakharov–Kuznetsov equation. A lattice Boltzmann model is constructed. In numerical experiments, the propagation of the single solitary wave and the collision of double solitary waves are simulated. The results with different parameters are investigated and compared.     

1-soliton solution of the generalized Zakharov–Kuznetsov modified equal width equation

This paper obtains the solitary wave solution of the generalized Zakharov–Kuznetsov modified equal width equation. The solitary wave ansatz method is used to carry out the integration of this equation. A couple of conserved quantities are calculated. The domain restriction is identified for the power law nonlinearity parameter.     

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.     

Numerical solution of the Burgers’ equation by local discontinuous Galerkin method

In this paper, the Burgers’ equation is transformed into the linear diffusion equation by using the Hopf–Cole transformation. The obtained linear diffusion equation is discretized in space by the local discontinuous Galerkin method. The temporal discretization is accomplished by the total variation diminishing Runge–Kutta method. Numerical solutions are compared with the exact solution and the numerical solutions obtained by Adomian’s decomposition method, finite difference method, B-spline finite element method and boundary element method. The results show that the local discontinuous Galerkin method is one of the most efficient methods for solving the Burgers’ equation. Even with small viscosity coefficient, it can get the satisfied solution.     

Exponential collocation methods based on continuous finite element approximations for efficiently solving the cubic Schrödinger equation

In this paper we derive and analyze new exponential collocation methods to efficiently solve the cubic Schrödinger Cauchy problem on a d -dimensional torus. The novel methods are formulated based on continuous time finite element approximations in a generalized function space. Energy preservation is a key feature of the cubic Schrödinger equation. It is proved that the novel methods can be of arbitrarily high order which exactly or approximately preserve the continuous energy of the original continuous system. The existence and uniqueness, regularity, and convergence of the new methods are studied in detail. Two practical exponential collocation methods are constructed, and three illustrative numerical experiments are included. The numerical results show the remarkable accuracy and efficiency of the new methods in comparison with existing numerical methods in the literature.     

Error estimates of the spectral collocation method for the Ginzburg–Landau equation coupled with the Benjamin–Bona–Mahony equation

In this paper, we consider the spectral collocation method for the Ginzburg–Landau equation coupled with the Benjamin–Bona–Mahony equation. Semidiscrete and fully discrete spectral collocation schemes are given. In the fully discrete case, a three-level spectral collocation scheme is considered. An energy estimation method is used to obtain error estimates for the approximate solutions. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 57, Suzdal Conference–2006, Part 3, 2008.     

An extension of a data assimilation method based on the application of the Fokker–Planck equation

A data assimilation method based on the Kalman filter theory and on the Fokker–Planck equation is extended to assimilate Atlantic Ocean data into a new version of the well-known Modular Ocean Model (MOM_3) from NOAA/GFDL. This extension enables assimilation of non-uniformly distributed data in space and time. Numerical experiments with Levitus atlas data are carried out with the ocean model configured at a low resolution. Some results of these experiments as well as other possible expansions are discussed.     

Numerical solutions of Burgers–Huxley equation by exact finite difference and NSFD schemes

In this paper, numerical solution of the Burgers–Huxley (BH) equation is presented based on the nonstandard finite difference (NSFD) scheme. At first, two exact finite difference schemes for BH equation obtained. Moreover an NSFD scheme is presented for this equation. The positivity, boundedness and local truncation error of the scheme are discussed. Finally, the numerical results of the proposed method with those of some available methods compared.     

A Note on the analytic solutions of the Camassa–Holm equation

In this Note we are concerned with the well-posedness of the Camassa–Holm equation in analytic function spaces. Using the Abstract Cauchy–Kowalewski Theorem we prove that the Camassa–Holm equation admits, locally in time, a unique analytic solution. Moreover, if the initial data is real analytic, belongs to $H^s(\mathbb{R})$ with $s>3/2$, $6u_0{6}_{L^1}<\infty$ and $u_0?u_{0xx}$ does not change sign, we prove that the solution stays analytic globally in time. To cite this article: M.C. Lombardo et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Numerical computation of the finite-genus solutions of the Korteweg–de Vries equation via Riemann–Hilbert problems

In this letter we describe how to compute the finite-genus solutions of the Korteweg–de Vries equation using a Riemann–Hilbert problem that is satisfied by the Baker–Akhiezer function corresponding to a Schrödinger operator with finite-gap spectrum. The recovery of the corresponding finite-genus solution is performed using the asymptotics of the Baker–Akhiezer function. This method has the benefit that the space and time dependence of the Baker–Akhiezer function appear in an explicit, linear and computable way. We make use of recent advances in the numerical solution of Riemann–Hilbert problems to produce an efficient and uniformly accurate numerical method for computing all finite-genus solutions of the KdV equation.     

Convergence of the method of fundamental solutions for Poisson’s equation on the unit sphere

Solutions of boundary value problems of the Laplace equation on the unit sphere are constructed by using the fundamental solution

Exact traveling wave solutions for the Benjamin–Bona–Mahony equation by improved Fan sub-equation method

This paper combines the bifurcation theory of dynamical systems and the Fan sub-equation method to improve the Fan sub-equation method for solving the BBM equation. Periodic solutions, kink solutions and solitary solutions are formally derived in a general form. This method is straightforward and concise, and it can also be applied to other nonlinear evolution equations.     

Classification of radial solutions to the Emden–Fowler equation on the hyperbolic space

We study the Emden–Fowler equation ?Δu = |u| ^p?1 u on the hyperbolic space ${{\mathbb H}^n}$ . We are interested in radial solutions, namely solutions depending only on the geodesic distance from a given point. The critical exponent for such equation is p = (n + 2)/(n ? 2) as in the Euclidean setting, but the properties of the solutions show striking differences with the Euclidean case. While the papers (Bhakta and Sandeep, Poincaré Sobolev equations in the hyperbolic space, 2011; Mancini and Sandeep, Ann Sci Norm Sup Pisa Cl Sci 7(5):635–671, 2008) consider finite energy solutions, we shall deal here with infinite energy solutions and we determine the exact asymptotic behavior of wide classes of finite and infinite energy solutions.     

Numerical study of a particle separation method based on the Segré-Silberberg effect

Particles that are placed in a laminar pipe flow rotate and migrate transversally to a radial equilibrium position. This so called Segré-Silberberg effect is used in a new method for size separation of particles. The particles to be separated are placed in a pipe flow and subsequently enter an expansion chamber, where the flow is split and the particles are divided into two fractions. This paper reports the results of two-dimensional Euler-Lagrange simulations of the motion of neutrally-buoyant particles inside the expansion chamber. The simulation results agree well with experimental data on the separation and show that the Saffman force has a significant impact onto the particle trajectories. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Continuous on rays solutions of an equation of the Go?a?b-Schinzel type

Let X be a real linear space. We characterize continuous on rays solutions f,g:X→R of the equation f(x+g(x)y)=f(x)f(y). Our result refers to papers of J. Chudziak (2006) [14] and J. Brzd?k (2003) [11].     

A Legendre–Galerkin Chebyshev collocation method for the Burgers equation with a random perturbation on boundary condition

In the paper, we apply the generalized polynomial chaos expansion and spectral methods to the Burgers equation with a random perturbation on its left boundary condition. Firstly, the stochastic Galerkin method combined with the Legendre–Galerkin Chebyshev collocation scheme is adopted, which means that the original equation is transformed to the deterministic nonlinear equations by the stochastic Galerkin method and the Legendre–Galerkin Chebyshev collocation scheme is used to deal with the resulting nonlinear equations. Secondly, the stochastic Legendre–Galerkin Chebyshev collocation scheme is developed for solving the stochastic Burgers equation; that is, the stochastic Legendre–Galerkin method is used to discrete the random variable meanwhile the nonlinear term is interpolated through the Chebyshev–Gauss points. Then a set of deterministic linear equations can be obtained, which is in contrast to the other existing methods for the stochastic Burgers equation. The mean square convergence of the former method is analyzed. Numerical experiments are performed to show the effectiveness of our two methods. Both methods provide alternative approaches to deal with the stochastic differential equations with nonlinear terms.