The Lie-group shooting method for solving nonlinear singularly perturbed boundary value problems

A new computational method for solving the second-order nonlinear singularly perturbed boundary value problems (SPBVPs) is provided in this paper. In order to overcome a highly singular behavior very near to the boundary as being not easy to treat by numerical method, we adopt a coordinate transformation from an x-domain to a t-domain via a rescaling technique, which can reduce the singularity within the boundary layer. Then, we construct a Lie-group shooting method (LGSM) to search a missing initial condition through the finding of a suitable value of a parameter r ∈ [0, 1]. Moreover, we can derive a closed-form formula to express the initial condition in terms of r, which can be determined properly by an accurate matching to the right-boundary condition. Numerical examples are examined, showing that the present approach is highly efficient and accurate.     

Swift-Hohenberg方程的打靶法

§ 1　IntroductionIn this paper we shall study the formation of spatially periodic patterns in extendedsystems described by Swift- Hohenberg equationut=ku - 1 +2x22 u - u3 ,k∈ R. (1.1)This equation was first proposed in 1976 by Swiftand Hohenberg[12 ] as a simple model forthe Rayleigh- B nard instability of roll waves.However,since then an effective m odel e-quation has been proved for a variety of system s in physics and mechanics.The Swift- Hohenberg equation has been studied a …     

Efficient Jacobi-Gauss collocation method for solving initial value problems of Bratu type

In this paper, we propose the shifted Jacobi-Gauss collocation spectral method for solving initial value problems of Bratu type, which is widely applicable in fuel ignition of the combustion theory and heat transfer. The spatial approximation is based on shifted Jacobi polynomials J _n ^(α,β) (x) with α, β ∈ (?1, ∞), x ∈ [0, 1] and n the polynomial degree. The shifted Jacobi-Gauss points are used as collocation nodes. Illustrative examples have been discussed to demonstrate the validity and applicability of the proposed technique. Comparing the numerical results of the proposed method with some well-known results show that the method is efficient and gives excellent numerical results.     

One-point pseudospectral collocation for the one-dimensional Bratu equation

The one-dimensional planar Bratu problem is u_xx + λ exp(u) = 0 subject to u(±1) = 0. Because there is an analytical solution, this problem has been widely used to test numerical and perturbative schemes. We show that over the entire lower branch, and most of the upper branch, the solution is well approximated by a parabola, u(x) ≈ u₀ (1 − x²) where u₀ is determined by collocation at a single point x = ξ. The collocation equation can be solved explicitly in terms of the Lambert W-function as u(0) ≈ −W(−λ(1 − ξ²)/2)/(1 − ξ²) where both real-valued branches of the W-function yield good approximations to the two branches of the Bratu function. We carefully analyze the consequences of the choice of ξ. We also analyze the rate of convergence of a series of even Chebyshev polynomials which extends the one-point approximation to arbitrary accuracy. The Bratu function is so smooth that it is actually poor for comparing methods because even a bad, inefficient algorithm is successful. It is, however, a solution so smooth that a numerical scheme (the collocation or pseudospectral method) yields an explicit, analytical approximation. We also fill some gaps in theory of the Bratu equation. We prove that the general solution can be written in terms of a single, parameter-free β(x) without knowledge of the explicit solution. The analytical solution can only be evaluated by solving a transcendental eigenrelation whose solution is not known explicitly. We give three overlapping perturbative approximations to the eigenrelation, allowing the analytical solution to be easily evaluated throughout the entire parameter space.     

Back-and-forth shooting method for solving two-point boundary-value problems

The paper proposes a special iterative method for a nonlinear TPBVP of the form (t)=f(t, x(t),p(t)), (t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.     

A shooting method for a nonlinear beam equation

We study the existence of solutions for a nonlinear fourth-order ODE with nonlinear boundary condition that arises in beam theory. Using a shooting type argument, we prove the existence of at least one solution of the problem.     

Convergence of the back-and-forth shooting method for solving two-point boundary-value problems

The back-and-forth shooting method of Orava and Lautala (Ref. 1) is considered. The method transforms a given boundary-value problem to a sequence of initial-value problems. The present paper studies the convergence properties of this sequence. A local convergence theorem is given, and the rate of convergence is found to be quadratic in sufficiently smooth cases. The necessary tools for this analysis concerning the Fréchet differentiability of certain mappings are given in the Appendix.     

Efficient shooting method for solving two point boundary value problems

We present an efficient shooting method for solving two point boundary value problems. The Adomian decomposition method will be utilized to obtain a series solution of the initial value problems involved. Numerical examples and comparison of the work of others will also be done.     

The King-Werner method for solving nonsymmetric algebraic Riccati equation

For the nonsymmetric algebraic Riccati equation arising from transport theory, we concern about solving its minimal positive solution. In [1], Lu transferred the equation into a vector form and pointed out that the minimal positive solution of the matrix equation could be obtained via computing that of the vector equation. In this paper, we use the King-Werner method to solve the minimal positive solution of the vector equation and give the convergence and error analysis of the method. Numerical tests show that the King-Werner method is feasible to determine the minimal positive solution of the vector equation.     

The modified simple equation method and the multiple exp-function method for solving nonlinear fractional Sharma-Tasso-Olver equation

In this article, the fractional derivatives in the sense of the modified Riemann-Liouville derivatives together with the modified simple equation method and the multiple exp-function method are employed for constructing the exact solutions and the solitary wave solutions for the nonlinear time fractional Sharma-Tasso- Olver equation. With help of Maple, we can get exact explicit 1-wave, 2-wave and 3-wave solutions, which include 1-soliton, 2-soliton and 3-soliton type solutions if we use the multiple exp-function method while we can get only exact explicit 1-wave solution including 1-soliton type solution if we use the modified simple equation method. Two cases with specific values of the involved parameters are plotted for each 2-wave and 3-wave solutions.     

The dual integral equation method for solving the heat conduction equation for an unbounded plate

The solution of the two-dimensional nonstationary heat conduction equation in axially symmetrical cylindrical coordinates for an unbounded plate is determined in this paper. A solution of the problem is given in the form of functional series, for which every term of a series represents an unknown function of a second kind integral equation. A new type of dual integral equations is used to solve a given boundary-value problem with the help of a Laplace transform and separation of variables.     

Legendre spectral method for the fractional Bratu problem

In this paper, the Legendre spectral collocation method (LSCM) is applied for the solution of the fractional Bratu's equation. It shows the high accuracy and low computational cost of the LSCM compared with some other numerical methods. The fractional Bratu differential equation is transformed into a nonlinear system of algebraic equations for the unknown Legendre coefficients and solved with some spectral collocation methods. Some illustrative examples are also given to show the validity and applicability of this method, and the obtained results are compared with the existing studies to highlight its high efficiency and neglectable error.     

A numerical method for solving the hyperbolic telegraph equation

Recently, it is found that telegraph equation is more suitable than ordinary diffusion equation in modelling reaction diffusion for such branches of sciences. In this article, we propose a numerical scheme to solve the one‐dimensional hyperbolic telegraph equation using collocation points and approximating the solution using thin plate splines radial basis function. The scheme works in a similar fashion as finite difference methods. The results of numerical experiments are presented, and are compared with analytical solutions to confirm the good accuracy of the presented scheme. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

A second-order accurate method for solving the eikonal equation

We present a numerical method for computing the signed distance to a piecewise-smooth surface defined as the zero set of a function. It is based on a marching method by Kim and a hybrid discretization of first- and second-order discretizations of the eikonal equation. If the solution is smooth at a point and at all of the points in the domain of dependence of that point, the solution is second-order accurate; otherwise, the method is first-order accurate, and computes the computes the correct entropy solution in the presence of kinks in the initial surface. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Conservative numerical method for solving the averaged Boltzmann equation

A method is proposed for averaging the Boltzmann kinetic equation with respect to transverse velocities. A system of two integro-differential equations for two desired functions depending only on the longitudinal velocity is derived. The gas particles are assumed to interact as absolutely hard spheres. The integrals in the equations are double. The reduction in the number of variables in the desired functions and the low multiplicity of the integrals ensure the computational efficiency of the averaged equations. A numerical method of discrete ordinates is developed that effectively solves the gas relaxation problem based on the averaged equations. The method is conservative, and the number of particles, momentum, and energy are conserved automatically at every time step.     

The contraction number of a multigrid method for solving the Poisson equation

Summary The treatment of a multigrid method in the framework of numerical analysis elucidates that regularity of the solution is not necessary for the convergence of the multigrid algorithm but only for fast convergence. For the linear equations which arise from the discretization of the Poisson equation, a convergence factor 0,5 is established independent of the shape of the domain and of the regularity of the solution.Dedicated to Professor Dr.Dr.h.c. Lothar Collatz on the occasion of his 70 th birthday     

FOURIER－CHEBYSHEV SPECTRAL METHOD FOR SOLVING THREE－DIMENSIONAL VORTICITY EQUATION

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