Boundary‐value problems for the third‐order loaded equation with noncharacteristic type‐change boundaries

The purpose of this paper is to establish unique solvability for a certain generalized boundary‐value problem for a loaded third‐order integro‐differential equation with variable coefficients. Moreover, the method of integral equations is applied to obtain an equation related to the Riemann‐Liouville operators.     

A Jacobi–Davidson method for two‐real‐parameter nonlinear eigenvalue problems arising from delay‐differential equations

The critical delays of a delay‐differential equation can be computed by solving a nonlinear two‐parameter eigenvalue problem. The solution of this two‐parameter problem can be translated to solving a quadratic eigenvalue problem of squared dimension. We present a structure preserving QR‐type method for solving such quadratic eigenvalue problem that only computes real‐valued critical delays; that is, complex critical delays, which have no physical meaning, are discarded. For large‐scale problems, we propose new correction equations for a Newton‐type or Jacobi–Davidson style method, which also forces real‐valued critical delays. We present three different equations: one real‐valued equation using a direct linear system solver, one complex valued equation using a direct linear system solver, and one Jacobi–Davidson style correction equation that is suitable for an iterative linear system solver. We show numerical examples for large‐scale problems arising from PDEs. Copyright © 2012 John Wiley & Sons, Ltd.     

Oscillation result for half‐linear dynamic equations on timescales and its consequences

We study oscillatory properties of half‐linear dynamic equations on timescales. Via the combination of the Riccati technique and an averaging method, we find the domain of oscillation for many equations. The presented main result is not the conversion of a known result from the theory of differential or difference equations, ie, we obtain new results for the timescales (for differential equations) and (for difference equations). Half‐linear equations generalize linear equations (in fact, they coincide with certain one‐dimensional PDEs with p‐Laplacian), but the main result is new also for linear differential and difference equations. The corresponding corollaries and examples are given as well.     

A multigrid compact finite difference method for solving the one‐dimensional nonlinear sine‐Gordon equation

The aim of this paper is to propose a multigrid method to obtain the numerical solution of the one‐dimensional nonlinear sine‐Gordon equation. The finite difference equations at all interior grid points form a large sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a compact finite difference scheme of fourth‐order for discretizing the spatial derivative and the standard second‐order central finite difference method for the time derivative. The proposed method uses the Richardson extrapolation method in time variable. The obtained system has been solved by V‐cycle multigrid (VMG) method, where the VMG method is used for solving the large sparse linear systems. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional sine‐Gordon equation. Copyright © 2014 John Wiley & Sons, Ltd.     

On order‐reducible sinc discretizations and block‐diagonal preconditioning methods for linear third‐order ordinary differential equations

By introducing a variable substitution, we transform the two‐point boundary value problem of a third‐order ordinary differential equation into a system of two second‐order ordinary differential equations (ODEs). We discretize this order‐reduced system of ODEs by both sinc‐collocation and sinc‐Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order‐reduced system of ODEs. The coefficient matrix of the linear system is of block two‐by‐two structure, and each of its blocks is a combination of Toeplitz and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block‐diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Computational methods of solving the boundary value problems for the loaded differential and Fredholm integro‐differential equations

The article presents a new general solution to a loaded differential equation and describes its properties. Solving a linear boundary value problem for loaded differential equation is reduced to the solving a system of linear algebraic equations with respect to the arbitrary vectors of general solution introduced. The system's coefficients and right sides are computed by solving the Cauchy problems for ordinary differential equations. Algorithms of constructing a new general solution and solving a linear boundary value problem for loaded differential equation are offered. Linear boundary value problem for the Fredholm integro‐differential equation is approximated by the linear boundary value problem for loaded differential equation. A mutual relationship between the qualitative properties of original and approximate problems is obtained, and the estimates for differences between their solutions are given. The paper proposes numerical and approximate methods of solving a linear boundary value problem for the Fredholm integro‐differential equation and examines their convergence, stability, and accuracy.     

On existence of solutions of differential‐difference equations

This paper applies Nevanlinna theory of value distribution to discuss existence of solutions of certain types of non‐linear differential‐difference equations such as (5) and (8) given in the succeeding paragraphs. Existence of solutions of differential equations and difference equations can be said to have been well studied, that of differential‐difference equations, on the other hand, have been paid little attention. Such mixed type equations have great significance in applications. This paper, in particular, generalizes the Rellich–Wittich‐type theorem and Malmquist‐type theorem about differential equations to the case of differential‐difference equations. Copyright © 2015 John Wiley & Sons, Ltd.     

An efficient numerical approach to solve a class of variable‐order fractional integro‐partial differential equations

The main purpose of this work is to investigate an initial boundary value problem related to a suitable class of variable order fractional integro‐partial differential equations with a weakly singular kernel. To discretize the problem in the time direction, a finite difference method will be used. Then, the Sinc‐collocation approach combined with the double exponential transformation is employed to solve the problem in each time level. The proposed numerical algorithm is completely described and the convergence analysis of the numerical solution is presented. Finally, some illustrative examples are given to demonstrate the pertinent features of the proposed algorithm.     

The use of cubic B‐spline scaling functions for solving the one‐dimensional hyperbolic equation with a nonlocal conservation condition

Problems for parabolic partial differential equations with nonlocal boundary conditions have been studied in many articles, but boundary value problems for hyperbolic partial differential equations have so far remained nearly uninvestigated. In this article a numerical technique is presented for the solution of a nonclassical problem for the one‐dimensional wave equation. This method uses the cubic B‐spline scaling functions. Some numerical results are reported to support our study. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

New predictor‐corrector scheme for solving nonlinear differential equations with Caputo‐Fabrizio operator

In this paper, we develop a new, simple, and accurate scheme to obtain approximate solution for nonlinear differential equation in the sense of Caputo‐Fabrizio operator. To derive this new predictor‐corrector scheme, which suits on Caputo‐Fabrizio operator, firstly, we obtain the corresponding initial value problem for the differential equation in the Caputo‐Fabrizio sense. Hence, by fractional Euler method and fractional trapeziodal rule, we obtain the predictor formula as well as corrector formula. Error analysis for this new method is derived. To test the validity and simplicity of this method, some illustrative examples for nonlinear differential equations are solved.     

Fourth‐order compact scheme for the one‐dimensional sine‐Gordon equation

Finite difference scheme to the generalized one‐dimensional sine‐Gordon equation is considered in this paper. After approximating the second order derivative in the space variable by the compact finite difference, we transform the sine‐Gordon equation into an initial‐value problem of a second‐order ordinary differential equation. Then Padé approximant is used to approximate the time derivatives. The resulting fully discrete nonlinear finite‐difference equation is solved by a predictor‐corrector scheme. Both Dirichlet and Neumann boundary conditions are considered in our proposed algorithm. Stability analysis and error estimate are given for homogeneous Dirichlet boundary value problems using energy method. Numerical results are given to verify the condition for stability and convergence and to examine the accuracy and efficiency of the proposed algorithm. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Bernstein Ritz‐Galerkin method for solving an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

In this article, the Ritz‐Galerkin method in Bernstein polynomial basis is implemented to give an approximate solution of a hyperbolic partial differential equation with an integral condition. We will deal here with a type of nonlocal boundary value problem, that is, the solution of a hyperbolic partial differential equation with a nonlocal boundary specification. The nonlocal conditions arise mainly when the data on the boundary cannot be measured directly. The properties of Bernstein polynomial and Ritz‐Galerkin method are first presented, then Ritz‐Galerkin method is used to reduce the given hyperbolic partial differential equation to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique presented in this article. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2010     

Strang‐type preconditioners applied to ordinary and neutral differential‐algebraic equations

This paper deals with boundary‐value methods (BVMs) for ordinary and neutral differential‐algebraic equations. Different from what has been done in Lei and Jin (Lecture Notes in Computer Science, vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large‐sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang‐type block‐circulant preconditioners are suggested: one is for ordinary differential‐algebraic equations (ODAEs), and the other is for neutral differential‐algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.     

A numerical approach for solving weakly singular partial integro‐differential equations via two‐dimensional‐orthonormal Bernstein polynomials with the convergence analysis

In this paper, we develop an efficient matrix method based on two‐dimensional orthonormal Bernstein polynomials (2D‐OBPs) to provide approximate solution of linear and nonlinear weakly singular partial integro‐differential equations (PIDEs). First, we approximate all functions involved in the considerable problem via 2D‐OBPs. Then, by using the operational matrices of integration, differentiation, and product, the solution of Volterra singular PIDEs is transformed to the solution of a linear or nonlinear system of algebraic equations which can be solved via some suitable numerical methods. With a small number of bases, we can find a reasonable approximate solution. Moreover, we establish some useful theorems for discussing convergence analysis and obtaining an error estimate associated with the proposed method. Finally, we solve some illustrative examples by employing the presented method to show the validity, efficiency, high accuracy, and applicability of the proposed technique.     

Determination of a control function in three‐dimensional parabolic equations by Legendre pseudospectral method

A Legendre pseudospectral method is proposed for solving approximately an inverse problem of determining an unknown control parameter p(t) which is the coefficient of the solution u(x, y, z, t) in a diffusion equation in a three‐dimensional region. The diffusion equation is to be solved subject to suitably prescribed initial‐boundary conditions. The presence of the unknown coefficient p(t) requires an extra condition. This extra condition considered as the integral overspecification over the spacial domain. For discretizing the problem, after homogenization of the boundary conditions, we apply the Legendre pseudospectral method in a matrix based manner. As a results a system of nonlinear differential algebraic equations is generated. Then by using suitable transformation, the problem will be converted to a homogeneous time varying system of linear ordinary differential equations. Also a pseudospectral method for efficient solving of the resulted system of ordinary differential equations is proposed. The solution of this system gives the approximation to values of u and p. The matrix based structure of the present method makes it easy to implement. Numerical experiments are presented to demonstrate the accuracy and the efficiency of the proposed computational procedure. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 74‐93, 2012     

D'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon equation

In this paper, we construct a weakly‐nonlinear d'Alembert‐type solution of the Cauchy problem for the Boussinesq‐Klein‐Gordon (BKG) equation. Similarly to our earlier work based on the use of spatial Fourier series, we consider the problem in the class of periodic functions on an interval of finite length (including the case of localized solutions on a large interval), and work with the nonlinear partial differential equation with variable coefficients describing the deviation from the oscillating mean value. Unlike our earlier paper, here we develop a novel multiple‐scales procedure involving fast characteristic variables and two slow time scales and averaging with respect to the spatial variable at a constant value of one or another characteristic variable, which allows us to construct an explicit and compact d'Alembert‐type solution of the nonlinear problem in terms of solutions of two Ostrovsky equations emerging at the leading order and describing the right‐ and left‐propagating waves. Validity of the constructed solution in the case when only the first initial condition for the BKG equation may have nonzero mean value follows from our earlier results, and is illustrated numerically for a number of instructive examples, both for periodic solutions on a finite interval, and localized solutions on a large interval. We also outline an extension of the procedure to the general case, when both initial conditions may have nonzero mean values. Importantly, in all cases, the initial conditions for the leading‐order Ostrovsky equations by construction have zero mean, while initial conditions for the BKG equation may have nonzero mean values.     

On the stability of alternating‐direction explicit methods for advection‐diffusion equations

Alternating‐Direction Explicit (A.D.E.) finite‐difference methods make use of two approximations that are implemented for computations proceeding in alternating directions, e.g., from left to right and from right to left, with each approximation being explicit in its respective direction of computation. Stable A.D.E. schemes for solving the linear parabolic partial differential equations that model heat diffusion are well‐known, as are stable A.D.E. schemes for solving the first‐order equations of fluid advection. Several of these are combined here to derive A.D.E. schemes for solving time‐dependent advection‐diffusion equations, and their stability characteristics are discussed. In each case, it is found that it is the advection term that limits the stability of the scheme. The most stable of the combinations presented comprises an unconditionally stable approximation for computations carried out in the direction of advection of the system, from left to right in this case, and a conditionally stable approximation for computations proceeding in the opposite direction. To illustrate the application of the methods and verify the stability conditions, they are applied to some quasi‐linear one‐dimensional advection‐diffusion problems. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Mixed two‐grid finite difference methods for solving one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations

The aim of this paper is to propose mixed two‐grid finite difference methods to obtain the numerical solution of the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. The finite difference equations at all interior grid points form a large‐sparse linear system, which needs to be solved efficiently. The solution cost of this sparse linear system usually dominates the total cost of solving the discretized partial differential equation. The proposed method is based on applying a family of finite difference methods for discretizing the spatial and time derivatives. The obtained system has been solved by two‐grid method, where the two‐grid method is used for solving the large‐sparse linear systems. Also, in the proposed method, the spectral radius with local Fourier analysis is calculated for different values of h and Δt. The numerical examples show the efficiency of this algorithm for solving the one‐dimensional and two‐dimensional Fitzhugh–Nagumo equations. Copyright © 2016 John Wiley & Sons, Ltd.     

On the solution of an initial‐boundary value problem that combines Neumann and integral condition for the wave equation

Numerical solution of hyperbolic partial differential equation with an integral condition continues to be a major research area with widespread applications in modern physics and technology. Many physical phenomena are modeled by nonclassical hyperbolic boundary value problems with nonlocal boundary conditions. In place of the classical specification of boundary data, we impose a nonlocal boundary condition. Partial differential equations with nonlocal boundary specifications have received much attention in last 20 years. However, most of the articles were directed to the second‐order parabolic equation, particularly to heat conduction equation. We will deal here with new type of nonlocal boundary value problem that is the solution of hyperbolic partial differential equations with nonlocal boundary specifications. These nonlocal conditions arise mainly when the data on the boundary can not be measured directly. Several finite difference methods have been proposed for the numerical solution of this one‐dimensional nonclassic boundary value problem. These computational techniques are compared using the largest error terms in the resulting modified equivalent partial differential equation. Numerical results supporting theoretical expectations are given. Restrictions on using higher order computational techniques for the studied problem are discussed. Suitable references on various physical applications and the theoretical aspects of solutions are introduced at the end of this article. © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005