Compact Difference Scheme with High Accuracy for One-Dimensional Unsteady Quasi-Linear Biharmonic Problem of Second Kind: Application to Physical Problems

The present paper uses a new two-level implicit difference formula for the numerical study of one-dimensional unsteady biharmonic equation with appropriate initial and boundary conditions. The proposed difference scheme is second-order accurate in time and third-order accurate in space on non-uniform grid and in case of uniform mesh, it is of order two in time and four in space. The approximate solutions are computed without using any transformation and linearization. The simplicity of the proposed scheme lies in its three-point spatial discretization that yields block tri-diagonal matrix structure without the use of any fictitious nodes for handling the boundary conditions. The proposed scheme is directly applicable to singular problems, which is the main utility of our work. The method is shown to be unconditionally stable for model linear problem for uniform mesh. The efficacy of the proposed approach has been tested on several physical problems, including the complex fourth-order nonlinear equations like Kuramoto–Sivashinsky equation and extended Fisher–Kolmogorov equation, where comparison is done with some earlier work. It is clear from numerical experiments that the obtained results are not only in good agreement with the exact solutions but also competent with the solutions derived in earlier research studies.     

On a finite difference scheme for Burgers’ equation

In this paper we study properties of numerical solutions of Burger’s equation. Burgers’ equation is reduced to the heat equation on which we apply the Douglas finite difference scheme. The method is shown to be unconditionally stable, fourth order accurate in space and second order accurate in time. Two test problems are used to validate the algorithm. Numerical solutions for various values of viscosity are calculated and it is concluded that the proposed method performs well.     

A novel efficient method for nonlinear boundary value problems

We present two new two-level compact implicit variable mesh numerical methods of order two in time and two in space, and of order two in time and three in space for the solution of 1D unsteady quasi-linear biharmonic problem subject to suitable initial and boundary conditions. The simplicity of the proposed methods lies in their three-point discretization without requiring any fictitious points for incorporating the boundary conditions. The derived methods are shown to be unconditionally stable for a model linear problem for uniform mesh. We also discuss how our formulation is able to handle linear singular problem and ensure that the developed numerical methods retain their orders and accuracy everywhere in the solution region. The proposed difference methods successfully works for the highly nonlinear Kuramoto-Sivashinsky equation. Many physical problems are solved to demonstrate the accuracy and efficiency of the proposed methods. The numerical results reveal that the obtained solutions not only approximate the exact solutions very well but are also much better than those available in earlier research studies.     

Existence of solutions to a discrete fourth order periodic boundary value problem

This paper is concerned with periodic boundary value problems for a fourth order nonlinear difference equation. Via variational methods and critical point theory, sufficient conditions are given for the existence of at least one solution, two solutions, and nonexistence of solutions. Our conditions do not involve the primitive function of the nonlinear term. Examples are provided to illustrate the applicability of the results.     

Highly accurate algorithms endowing with boundary functions for solving a nonlinear beam equation involving an integral term

In this paper, the problem of a nonlinear beam equation involving an integral term of the deformation energy, which is unknown before the solution, under different boundary conditions with simply supported, 2‐end fixed, and cantilevered is investigated. We transform the governing equation into an integral equation and then solve it by using the sinusoidal functions, which are chosen both as the test functions and the bases of numerical solution. Because of the orthogonality of the sinusoidal functions, we can find the expansion coefficients of the numerical solution that are given in closed form by using the Drazin inversion formula. Furthermore, we introduce the concept of fourth‐order and fifth‐order boundary functions in the solution bases, which can greatly raise the accuracy over 4 orders than that using the partial boundary functions. The iterative algorithms converge very fast to find the highly accurate numerical solutions of the nonlinear beam equation, which are confirmed by 6 numerical examples.     

Numerical solution of nonlinear Jaulent–Miodek and Whitham–Broer–Kaup equations

In this work we investigate the numerical solution of Jaulent–Miodek (JM) and Whitham–Broer–Kaup (WBK) equations. The proposed numerical schemes are based on the fourth-order time-stepping schemes in combination with discrete Fourier transform. We discretize the original partial differential equations (PDEs) with discrete Fourier transform in space and obtain a system of ordinary differential equations (ODEs) in Fourier space which will be solved with fourth order time-stepping methods. After transforming the equations to a system of ODEs, the linear operator in JM equation is diagonal but in WBK equation is not diagonal. However for WBK equation we can also implement the methods such as diagonal case which reduces the CPU time. Comparing numerical solutions with analytical solutions demonstrates that those methods are accurate and readily implemented.     

Fourth Order Symmetric Finite Difference Schemes for the Acoustic Wave Equation

We present an explicit, symmetric finite difference scheme for the acoustic wave equation on a rectangle with Neumann and/or Dirichlet boundary conditions. The scheme is fourth order accurate both in time and space. It is obtained by mass lumping of a finite element scheme. The accuracy and the difference approximations at the boundary are analyzed in terms of local and global errors. AMS subject classification (2000) 65M10     

Unconditionally stable modified methods for the solution of two‐ and three‐dimensional telegraphic equation with Robin boundary conditions

In this article, we discuss modified three level implicit difference methods of order two in time and four in space for the numerical solution of two‐ and three‐dimensional telegraphic equation with Robin boundary conditions. Ghost points are introduced to obtain fourth‐order approximations for boundary conditions. Matrix stability analysis is carried out to prove stability of the method for telegraphic equations in two and three dimensions with Neumann boundary conditions. Numerical experiments are carried out and the results are found to be better when compared with the results obtained by other existing methods.     

The asymptotics of the eigenvalues of a fourth order differential operator with summable coefficients

A fourth order differential operator with summable coefficients and some boundary conditions is considered. Asymptotics of solutions to a fourth order differential equation is studied. The equation for eigenvalues is also studied and an asymptotics of the eigenvalues of the considered boundary value problem is obtained.     

Efficient Implementation of a Spectral-Tau Method for a Class of Two-Point Boundary-Value and Initial-Boundary-Value Problems

A simple and efficient spectral method for solving the second, third order and fourth order elliptic equations with variable coefficients and nonlinear differential equations is presented. It is different from spectral-collocation method which leads to dense, ill-conditioned matrices. The spectral method in this paper solves for the coefficients of the solution in a Chebyshev series, leads to discrete systems with special structured matrices which can be factorized and solved efficiently. We also extend the method to boundary value problems in two space dimensions and solve 2-D separable equation with variable coefficients. As an application, we solve Cahn-Hilliard equation iteratively via first-order implicit time discretization scheme. Ample numerical results indicate that the proposed method is extremely accurate and efficient.     

Solving singular second order three-point boundary value problems using reproducing kernel Hilbert space method

This paper investigates the numerical solutions of singular second order three-point boundary value problems using reproducing kernel Hilbert space method. It is a relatively new analytical technique. The solution obtained by using the method takes the form of a convergent series with easily computable components. However, the reproducing kernel Hilbert space method cannot be used directly to solve a singular second order three-point boundary value problem, so we convert it into an equivalent integro-differential equation, which can be solved using reproducing kernel Hilbert space method. Four numerical examples are given to demonstrate the efficiency of the present method. The numerical results demonstrate that the method is quite accurate and efficient for singular second order three-point boundary value problems.     

非线性四阶常微分方程三点边值问题解的存在性

本文利用文献[1]、[2]的方法,讨论了非线性四阶常微分方程满足如下条件的三点边值问题解的存在性。对于非线性四阶常微子方程的边值问题,到目前已经有了一系列的研究。而本文所讨论的方程(*),具非线性边界条件(**)的边值问题解的存在性,则是上述工作未曾涉及的。本文主要定理的推论,方程(*)之具如下形式的边界条件的边值问题解的存在性,以前的工作也未涉及。     

A high order accurate numerical method for solving two‐dimensional dual‐phase‐lagging equation with temperature jump boundary condition in nanoheat conduction

Dual‐phase‐lagging (DPL) equation with temperature jump boundary condition (Robin's boundary condition) shows promising for analyzing nanoheat conduction. For solving it, development of higher‐order accurate and unconditionally stable (no restriction on the mesh ratio) numerical schemes is important. Because the grid size may be very small at nanoscale, using a higher‐order accurate scheme will allow us to choose a relative coarse grid and obtain a reasonable solution. For this purpose, recently we have presented a higher‐order accurate and unconditionally stable compact finite difference scheme for solving one‐dimensional DPL equation with temperature jump boundary condition. In this article, we extend our study to a two‐dimensional case and develop a fourth‐order accurate compact finite difference method in space coupled with the Crank–Nicolson method in time, where the Robin's boundary condition is approximated using a third‐order accurate compact method. The overall scheme is proved to be unconditionally stable and convergent with the convergence rate of fourth‐order in space and second‐order in time. Numerical errors and convergence rates of the solution are tested by two examples. Numerical results coincide with the theoretical analysis. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1742–1768, 2015     

A new high accuracy finite difference discretization for the solution of 2D nonlinear biharmonic equations using coupled approach

In this article, using coupled approach, we discuss fourth order finite difference approximation for the solution of two dimensional nonlinear biharmonic partial differential equations on a 9‐point compact stencil. The solutions of unknown variable and its Laplacian are obtained at each internal grid points. This discretization allows us to use the Dirichlet boundary conditions only and there is no need to discretize the derivative boundary conditions. We require only system of two equations to obtain the solution and its Laplacian. The proposed fourth order method is used to solve a set of test problems and produce high accuracy numerical solutions. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

A Novel Numerical Simulations for Fornberg-Whitham and Modified Fornberg-Whitham Equations with Nonhomogeneous Boundary Conditions

In this study, the numerical solutions of the Fornberg-Whitham (FW) equation modeling the qualitative behavior of wave refraction and the modified Fornberg-Whitham (mFW) equation describing the solitary wave and peakon waves with a discontinuous first derivative at the peak have been obtained. To obtain numerical results, the collocation finite element method has been combined with quintic B-spline bases. Although there are solutions to these equations by semi-analytical and analytical methods in the literature, there are very few studies using numerical methods. The stability analysis of the applied method is examined by the von-Neumann Fourier series method. We have considered four test problems with nonhomogeneous boundary conditions that have analytical solutions to show the performance of the method. The numerical results of the two problems are compared with some studies in the literature. Additionally, peakon wave solutions and some new numerical results of the mFW equation, which are not available in the literature, are given in the last two problems. No comparison has been made since there are no numerical results in the literature for the last two problems. The error norms $L_{2}$ and $L_{\infty }$ are calculated to demonstrate the presented numerical scheme''s accuracy and efficiency. The advantage of the scheme is that it produces accurate and reliable solutions even for modest values of space and time step lengths, rather than small values that cause excessive data storage in the computation process. In general, large step lengths in the space and time directions result in smaller matrices. This means less storage on the computer and results in faster outcomes. In addition, the present method gives more accurate results than some methods given in the literature.     

On Integral Representation Formulae for Biharmonic Functions on the Ball

Biharmonic functions are solutions of the fourth order partial differential equation ΔΔu = 0. A simple method is proposed for deriving integral representation formulae for these functions u on the n‐dimensional ball. Poisson‐type representations in the setting of Hardy Spaces are obtained for biharmonic functions subject to Dirichlet, Riquier and other boundary conditions. The approach exploits algebraic properties of a first order partial differential operator and its resolvent.     

On the convergence of a fourth order evolution equation to the Allen-Cahn equation

We construct special sequences of solutions to a fourth order nonlinear parabolic equation of Cahn-Hilliard/Allen-Cahn type, converging to the second order Allen-Cahn equation. We consider the evolution equation without boundary, as well as the stationary case on domains with Dirichlet boundary conditions. The proofs exploit the equivalence of the fourth order equation with a system of two second order elliptic equations with “good signs”.     

The use of compact boundary value method for the solution of two-dimensional Schrödinger equation

In this paper, a high-order and accurate method is proposed for solving the unsteady two-dimensional Schrödinger equation. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives and a boundary value method of fourth-order for the time integration of the resulting linear system of ordinary differential equations. The proposed method has fourth-order accuracy in both space and time variables. Moreover this method is unconditionally stable due to the favorable stability property of boundary value methods. The results of numerical experiments are compared with analytical solutions and with those provided by other methods in the literature. These results show that the combination of a compact finite difference approximation of fourth-order and a fourth-order boundary value method gives an efficient algorithm for solving the two dimensional Schrödinger equation.     

A numerical algorithm for solving a class of linear nonlocal boundary value problems

In order to solve a class of linear nonlocal boundary value problems, a new reproducing kernel space satisfying nonlocal conditions is constructed carefully. This makes it easy to solve the problems. Furthermore, the exact solutions of the problems can be expressed in series form. The numerical results demonstrate that the new method is quite accurate and efficient for solving fourth-order nonlocal boundary value problems.