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This paper deals with the numerical approximation of semilinear parabolic stochastic partial differential equation (SPDE) driven simultaneously by Gaussian noise and Poisson random measure, more realistic in modeling real world phenomena. The SPDE is discretized in space with the standard finite element method and in time with the linear implicit Euler method or an exponential integrator, more efficient and stable for stiff problems. We prove the strong convergence of the fully discrete schemes toward the mild solution. The results reveal how convergence orders depend on the regularity of the noise and the initial data. In addition, we exceed the classical orders 1 ∕ 2 in time and 1 in space achieved in the literature when dealing with SPDE driven by Poisson measure with less regularity assumptions on the nonlinear drift function. In particular, for trace class multiplicative Gaussian noise we achieve convergence order O ( h 2 + Δ t 1 ∕ 2 ) . For additive trace class Gaussian noise and an appropriate jump function, we achieve convergence order O ( h 2 + Δ t ) . Numerical experiments to sustain the theoretical results are provided.

Nonlocality and spatial heterogeneity of many practical systems have made fractional differential equations very useful tools in Science and Engineering. However, solving these type of models is computationally demanding. In this paper, we propose an exponential integrator method for space fractional models as an attractive and easy-to-code alternative for other existing second-order exponential integrator methods. This scheme is based on using a real distinct poles discretization for the underlying matrix exponentials. One of the major benefits of the proposed scheme is that the algorithm could be easily implemented in parallel to take advantage of multiple processors for increased computational efficiency. The scheme is established to be second-order convergent; and proven to be robust for nonlinear space fractional reaction–diffusion problems involving non-smooth initial data. Our approach is exhibited by solving a system of two-dimensional problems which exhibits pattern formation and has applications in cell-division. Empirically, super-diffusion processes are displayed by investigating the effect of the fractional power of the underlying Laplacian operator on the pattern formation found in these models. Furthermore, the superiority of our method over competing second order ETD schemes, BDF2 scheme, and IMEX schemes is demonstrated.

Fractional partial differential equations (PDEs) provide a powerful and flexible tool for modeling challenging phenomena including anomalous diffusion processes and long-range spatial interactions, which cannot be modeled accurately by classical second-order diffusion equations. However, numerical methods for space-fractional PDEs usually generate dense or full stiffness matrices, for which a direct solver requires O ( N 3 ) computations per time step and O ( N 2 ) memory, where N is the number of unknowns. The significant computational work and memory requirement of the numerical methods makes a realistic numerical modeling of three-dimensional space-fractional diffusion equations computationally intractable. Fast numerical methods were previously developed for space-fractional PDEs on multidimensional rectangular domains, without resorting to lossy compression, but rather, via the exploration of the tensor-product form of the Toeplitz-like decompositions of the stiffness matrices. In this paper we develop a fast finite difference method for distributed-order space-fractional PDEs on a general convex domain in multiple space dimensions. The fast method has an optimal order storage requirement and almost linear computational complexity, without any lossy compression. Numerical experiments show the utility of the method.

In this paper, we established some sufficient conditions for oscillation of solutions of a class of impulsive partial fractional differential equations with forcing term subject to Robin and Dirichlet boundary conditions by using differential inequality method. As an application, we included an example to illustrate the main result.

The aim of this paper is to introduce an efficient numerical algorithm for the solution of stochastic time fractional stiff partial differential equations (PDEs). The time fractional derivative is described in the Caputo sense. By applying polynomial chaos (PC) expansion to discretize the random variable, a coupled time fractional deterministic system of PDEs is obtained. For the resulting system of time fractional PDEs we apply the Fourier spectral method to discretize the spatial variable, and use the fast Fourier transform (FFT) during the computation. Then, we use an exponential integrator method to overcome the stability issues due to the stiffness in the resulting time fractional semi-discrete system. The considered models have two challenging parameters, fractional order of equations and noise amplitude. In addition to exponential integrator, we also implemented a predictor–corrector method of Adams–Bashforth–Moulton type. We include numerical results of applying the developed method to stochastic time fractional Burgers and Kuramoto–Sivashinsky (KS) equations, for various values of noise intensity. A comparison of performance of the proposed scheme with fractional Adams method is also reported which confirms the efficiency and applicability of the proposed method based on the exponential integrator scheme.

In this paper we present numerical methods for computing the matrix functions arising in exponential integrators. The matrix functions are the linear combination of the form ź j = 0 p ź j ( A ) b j , where ź j is related to exponential function. By reducing the corresponding systems of linear ODEs with polynomial inhomogeneity based on block Krylov subspace methods, two effective exponentials of low dimension matrix are derived to approximate the objective functions. Some useful a posteriori error estimates are established. The algorithms can be combined with time-stepping strategy to satisfy the accuracy requirements under a moderate dimension of the block Krylov subspace. Several numerical experiments are presented to demonstrate the efficiency of the methods.

We consider the numerical approximation of a general second order semi-linear parabolic partial differential equation. Equations of this type arise in many contexts, such as transport in porous media. Using the finite volume with two-point flux approximation on regular mesh combined with exponential time differencing of order one (ETD1) for temporal discretization, we derive the L 2 estimate under the assumption that the non linear term is locally Lipschitz. Numerical simulations to sustain the theoretical results are provided.

Existence and uniqueness of solutions of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations are proved using Banach fixed-point theorem. An iterative scheme is derived and rigorous convergence analysis of this scheme and an error estimate are included as well. Several numerical examples for high dimensional initial value problem for heat and wave type partial differential equations are presented to demonstrate reliability and performance of proposed iterative scheme.

We present a new two-level compact implicit numerical scheme based on off-step discretization for the solution of the system of three space dimensional quasi-linear parabolic partial differential equations subject to suitable initial and boundary conditions. Further, we derive the estimates of first order space derivatives of the solution. The proposed methods are fourth order accurate in space and second order accurate in time, and involve 19-spatial grid points of a single compact cell. Also, we develop the alternating direction implicit (ADI) scheme for a linear parabolic equation with variable coefficients, which is shown to be unconditionally stable for the heat equation in polar coordinates. The methods so proposed do not require any modification when applied to the singular problems at the point of singularity unlike the numerical scheme proposed earlier in Mohanty and Jain (1994), and Mohanty (1997, 2003). The proposed methods are directly applicable to parabolic equations with singular coefficients. This is the main highlight of our work. The method successfully works for the Navier-Stokes equations of motion in polar coordinates. Many physical problems are solved to illustrate the accuracy of the proposed methods.

We investigate the strong convergence of a full discretization for the stochastic elastic equation driven by multiplicative noise. Finite element method is used for the spatial approximation and an exponential integrator scheme is proposed for the temporal discretization. We show that the fully discrete scheme strongly converges to the mild solution of the continuous problem, with precise convergence rates given. As shown in our result, the exponential integrator scheme achieves a higher strong convergence rate than the backward Euler method does in the literature.

In this paper, a numerical method for solving the linear initial problems for partial differential equations with constant coefficients and analytic initial conditions in two and three independent variables is presented. The technique is based upon the Taylor series expansion. Properties and the operational matrices for partial derivatives for the Taylor series in two and three variables are first presented. These properties are then used to reduce the solution of partial differential equations in two and three independent variables to a system of algebraic equations. The procedure can be extended to linear partial differential equations with more independent variables. The Taylor series may not converge if the solution is not analytic in the whole domain, however, the present method can be applied to boundary-value problems for linear partial differential equations, when the solution is analytic in the interior of the domain and also on some open subsets for each distinct part of the boundary. The method is computationally very attractive and applications are demonstrated through illustrative examples.

The numerical solution of linear time-invariant systems of fractional order is investigated. We construct a family of exponential integrators of Adams type possessing good convergence and stability properties. The methods are devised in order to keep at a suitable level, the computational effort necessary to solve problems of large size. Numerical experiments are provided to validate the theoretical results; the effectiveness of the proposed approach is tested and compared to some other classical methods.

We present a new class of exponential integrators for ordinary differential equations: locally exact modifications of known numerical schemes. Local exactness means that they preserve the linearization of the original system at every point. In particular, locally exact integrators preserve all fixed points and are A-stable. We apply this approach to popular schemes including Euler schemes, implicit midpoint rule and trapezoidal rule. We found locally exact modifications of discrete gradient schemes (for symmetric discrete gradients and coordinate increment discrete gradients) preserving their main geometric property: exact conservation of the energy integral (for arbitrary multidimensional Hamiltonian systems in canonical coordinates). Numerical experiments for a 2-dimensional anharmonic oscillator show that locally exact schemes have very good accuracy in the neighbourhood of stable equilibrium, much higher than suggested by the order of new schemes (locally exact modification sometimes increases the order but in many cases leaves it unchanged)., 28 pages plus 6 figures, Computers & Mathematics with Applications, 2013. arXiv admin note: substantial text overlap with arXiv:1101.0578

For the time integration of semilinear systems of differential equations, a class of multiderivative exponential integrators is considered. The methods are based on a Taylor series expansion of the semilinearity about the numerical solution, the required derivatives are computed by automatic differentiation. Inserting these derivatives into the variation-of-constants formula results in an exponential integrator which requires the action of the exponential of an augmented Jacobian only. The convergence properties of such exponential integrators are analyzed, and potential sources of numerical instabilities are identified. In particular, it is shown that local linearization gives rise to better stability for stiff problems. A number of numerical experiments illustrate the theoretical results.

In this paper, a spectral-element discontinuous Galerkin (SEDG) lattice Boltzmann discretization and an exponential time-marching scheme are used to study the flow field past two circular cylinders in tandem arrangement. The basic idea is to discretize the streaming step of the lattice Boltzmann equation by using the SEDG method to get a system of ordinary differential equations (ODEs) whose exact solutions are expressed by using a large matrix exponential. The approximate solution of the resulting ODEs are obtained from a projection method based on a Krylov subspace approximation. This approach allows us to approximate the matrix exponential of a very large and sparse matrix by using a matrix of much smaller dimension. The exponential time integration scheme is useful especially when computations are carried out at high Courant-Friedrichs-Lewy (CFL) numbers, where most explicit time-marching schemes are inaccurate. Simulations of flow were carried out for a circular cylinder at Re=20 and for two circular cylinders in tandem at Re=40 and a spacing of 2.5D, where D is the diameter of the cylinders. We compare our results with those from a fourth-order Runge-Kutta scheme that is restricted by the CFL number. In addition, important flow parameters such as the drag coefficients of the two cylinders and the wake length behind the rear cylinder were calculated by using the exponential time integration scheme. These results are compared with results from our simulation using the RK scheme and with existing benchmark results.

In this paper, we consider the numerical solution of the Riesz space fractional diffusion equation and advection–dispersion equation. First, a system of ordinary differential equations is obtained from the above equations with respect to the space variable by using the improved matrix transform method. Furthermore, we use the (2,2) Pade approximation to compute the exponential matrix in the analytic solution of the ordinary differential equation, and get two difference schemes. Second, using the matrix analysis method, we prove that the two difference schemes are unconditionally stable. Finally, some numerical results are given, which demonstrate the effectiveness of the two difference schemes.

In this paper, three numerical methods to solve Volterra integro-differential equations containing rational functions are discussed. The first one is the Differential Quadrature Method and, to the best knowledge of the author, it has never been applied to this kind of problem; the second one is a new version of the Iterative Differential Quadrature method, a method proposed by the author some years ago to solve problems in space–time domains, revised herein for the single space variable problem; the third one is a numerical Picard-like method, proposed herein to combine successive approximations with numerical integration. Stability and convergence of the second and the third method are discussed. The three methods have been applied to solve a real world problem in the field of the structural engineering and the numerical results compared.

In this paper, a numerical method which produces an approximate polynomial solution is presented for solving the high-order linear singular differential–difference equations. With the aid of Bessel polynomials and collocation points, this method converts the singular differential–difference equations into the matrix equation. The matrix equation corresponds to a system of linear equations with the unknown Bessel coefficients. This method gives the analytic solutions when the exact solutions are polynomials. Finally, some experiments and their numerical solutions are given; by comparing the numerical results obtained from the other methods, we show the high accuracy and efficiency of the proposed method. All of the numerical computations have been performed on a PC using some programs written in MATLAB v7.6.0 (R2008a).

In this paper, we develop a framework to obtain approximate numerical solutions to ordinary differential equations (ODEs) involving fractional order derivatives using Legendre wavelet approximations. The properties of Legendre wavelets are first presented. These properties are then utilized to reduce the fractional ordinary differential equations (FODEs) to the solution of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of the technique. Results show that this technique can solve the linear and nonlinear fractional ordinary differential equations with negligible error compared to the exact solution.

The response to static loads of plane inhomogeneous anisotropic bodies made of linear viscoelastic materials is investigated. Multi-parameter differential viscoelastic constitutive equations are employed, which are generalized using fractional order time derivatives. The governing equations, which are derived by considering the equilibrium of the plane body element, are two coupled linear fractional evolution partial differential equations in terms of the displacement components. Using the Analog Equation Method (AEM) in conjunction with the Boundary Element Method (BEM) these equations are transformed into a system of multi-term ordinary fractional differential equations (FDEs), which are solved using a numerical method for FDEs developed recently by Katsikadelis. Numerical examples are presented, which not only demonstrate the efficiency of the solution procedure and validate its accuracy, but also permit a better understanding of the response of plane bodies described by different viscoelastic models.

This paper proposes a symmetry–iteration hybrid algorithm for solving boundary value problems for partial differential equations. First, the multi-parameter symmetry is used to reduce the problem studied to a simpler initial value problem for ordinary differential equations. Then the variational iteration method is employed to obtain its solution. The results reveal that the proposed method is very effective and can be applied for other nonlinear problems.

A non-standard finite difference scheme is developed to solve the linear partial differential equations with time- and space-fractional derivatives. The Grunwald–Letnikov method is used to approximate the fractional derivatives. Numerical illustrations that include the linear inhomogeneous time-fractional equation, linear space-fractional telegraph equation, linear inhomogeneous fractional Burgers equation and the fractional wave equation are investigated to show the pertinent features of the technique. Numerical results are presented graphically and reveal that the non-standard finite difference scheme is very effective and convenient for solving linear partial differential equations of fractional order.

Based on a transformed Painlevé property and the variable separated ODE method, a function transformation method is proposed to search exact solutions to some partial differential equations (PDEs) with hyperbolic or exponential functions. The new approach provides a more systematical and convenient handling of the solution process for the nonlinear equations. Its key point is to eradicate the hyperbolic or exponential terms by a transformed Painlevé property and reduce the given PDEs to a variable-coefficient ordinary differential equations, then we look for solutions to the resulting equations by some methods. As an application, exact solutions for a generalized sinh-Gordon equation are formally derived.

In this paper, the initial value problem is discussed for a class of fractional neutral functional differential equations and the criteria on existence are obtained.

Asymptotic equilibria of linear integro-differential equations and asymptotic relations between solutions of linear homogeneous impulsive differential equations and those of linear integro-differential equations are established. A new Gronwall-Bellman type lemma for integro-differential inequalities is proved. An example is given to demonstrate the validity of one of the results. © 2008 Elsevier Ltd. All rights reserved.

This paper is concerned with the numerical solution of delay integro-differential equations. The adaptation of linear multistep methods is considered. The emphasis is on the linear stability of numerical methods. It is shown that every A-stable, strongly 0-stable linear multistep method of Pouzet type can preserve the delay-independent stability of the underlying linear systems. In addition, some delay-dependent stability conditions for the stability of numerical methods are also given.

We consider the combustion equation as one of the candidates from the class of stiff ordinary differential equations. A solution over a length of time that is inversely proportional to δ>0 (where δ>0 is a small disturbance of the pre-ignition state) is sought. This problem has a transient at the midpoint of the integration interval. The solution changes from being non-stiff to stiff, and afterwards becomes non-stiff again. We provide its asymptotic and numerical solution obtained via a variety of methods. Comparisons are made for the numerical results which we obtain with the MATLAB ode solvers (ode45, ode15s and ode23s) and some nonstandard finite difference methods. Results corresponding to standard finite difference method are also presented. Furthermore, the discussion on these approaches along with the others, provides several open problems for new and young researchers.

In this paper, we introduce a new algorithm for applying the Adomian decomposition method to nonlinear differential and partial differential equations. The proposed algorithm does not require the explicit evaluation of Adomian polynomials to approximate the nonlinearity term. The numerical experiments show that the proposed algorithm is more accurate at larger values for time, though at the cost of more computations.

As a time discretization scheme for an ordinary differential equation with a stiff linear term, there is a class of methods that utilize the exponential or related functions of the coefficient matrix of the linear term. To implement these methods, we must compute a set of matrix functions called ''@f-function'', that includes the exponential itself, and it is important to compute these functions efficiently and accurately. In this paper, we consider the modified scaling and squaring method for the computation of @f-function. An algorithm based on Higham's method is defined, and the bounding parameter @q"m appropriate for @f-function is determined from an analysis of the truncation error under the assumption of the exact arithmetic. We also consider the propagation of the rounding error in the squaring process, and show that the error of @f-function is expected to be less than or roughly equal to that of the matrix exponential. Several evaluations are performed for famous test matrices, and the result shows that when the matrix exponential is computed accurately, the other @f-functions can also be obtained with the same level of accuracy.

This paper deals with the construction of numerical solutions of random initial value differential problems. The random Euler method is presented and the conditions for the mean square convergence are established. Numerical examples show that random Euler method gives good results even if the sufficient convergence conditions are not satisfied.

We study periodic boundary value problems relative to a general class of first-order functional differential equations. For this class of problems, we develop the monotone iterative technique. Our formulation is very general, including delay differential equations, functional differential equations with maxima and integro-differential equations, but the case where the operator defining the functional dependence is not necessarily Lipschitzian is also considered.

We find conditions for the solution to a first-order differential equations to be: bounded by an exponential function, uniformly stable, and uniformly asymptotically stable. Estimates are obtained using Liapunov functionals (energy functionals). Our assumptions are weaker than those in the references, and results are extended to systems of equations and to nonlinear equations.

The Taylor series method is one of the earliest analytic-numeric algorithms for approximate solution of initial value problems for ordinary differential equations. The main idea of the rehabilitation of this algorithms is based on the approximate calculation of higher derivatives using well-known finite-difference technique for the partial differential equations. The approximate solution is given as a piecewise polynomial function defined on the subintervals of the whole interval integration. This property offers different facility for adaptive error control. This paper describes several explicit Taylor series algorithms with numerical derivatives and their implicit extension and examines its consistency and stability properties. The implicit extension based on a collocation term added to the explicit truncated Taylor series and the approximate solution obtained as a continuously differentiable piecewise polynomials function. Some numerical test results is presented to prove the efficiency of these new-old algorithm.

A performance model is constructed for parallel iterative numerical methods under the assumption of a message-passing computing system. Arguments are given for the fact that the speedup of parallel iterative methods is mainly influenced by the speedup at one iterative step. Using the theoretical model, it is proved why explicit iterative methods for ordinary differential equations are inefficient in implementation on distributed memory multiprocessor systems. Numerical tests on parallel and distributed computing environments confirm the correctness of the theoretical model at least in the case of iterative methods for ordinary differential equations and time-dependent partial differential equations.

In recent years, many authors demonstrated the usefulness of fractional calculus operators in the derivation of (explicit) particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. In particular, by making use of the operators of fractional calculus based upon the Cauchy-Goursat integral formula, a certain family of nearly-simple harmonic vibration equations was considered systematically in a series of papers which appeared quite recently. The main object of the present sequel to all these earlier works is to investigate some relationships between such families of fractional differential equations and certain classes of ordinary differential equations. A preliminary interpretation of this family of nearly-simple harmonic vibration equations by means of an example involving linear electric circuits is also provided.

In recent years, many authors demonstrated the usefulness of fractional calculus operators in the derivation of (explicit) particular solutions of a number of linear ordinary and partial differential, equations of the second and higher orders. The main object of the present paper is to show how readily some recent contributions on this subject by several workers, involving various interesting classes of non-Fuchsian differential equations (including, for example, the Fukuhara and Tricomi equations and the celebrated Bessel and Whittaker equations), can be obtained (in a unified manner) by suitably applying some general theorems on (explicit) particular solutions of a certain family of linear ordinary fractional differintegral equations.

We study the application of moving mesh methods to a one-dimensional (time dependent) detonator delay element problem. We consider moving mesh methods based on the equidistribution principle derived by Huang et al. [1]. Adaptive mesh methods have been widely used recently to solve time dependent partial differential equations having large solution gradients. Significant improvements in accuracy and efficiency are achieved by adapting the nodes (mesh points) so that they are concentrated about areas of large solution variations. Each system of equations for the moving mesh methods is solved in conjunction with the detonator problem. In this paper, the system of ordinary differential equations that results (after discretising in space) is solved using the double precision version of the stiff ordinary differential equation solver DASSL. The numerical results clearly demonstrate that the moving mesh methods are capable of tracking the deflagration wave as it travels down the detonator delay element more accurately and more efficiently than a fixed mesh method.

In many recent works, one can find remarkable demonstrations of the usefulness of certain fractional calculus operators in the derivation of (explicit) particular solutions of a number of linear ordinary and partial differential equations of the second and higher orders. Our main objective in the present sequel to these earlier works, is to show how readily and systematically some recent contributions on this subject, involving linear ordinary and partial differential equations of order m (m ϵ N ) , can be obtained (in a unified manner) by suitably appealing to some general theorems on (explicit) particular solutions of a certain family of linear ordinary fractional differintegral equations.

In this work, we present a numerical method for approximating the solution of a linear differential problem. We formulate a general problem in a Banach space making use of a Schauder basis and the Tau method to approximate the load function and the solution of the differential problem. Finally, we offer a numerical example.

This paper is concerned with the numerical solution of functional-differential and functional equations which include functional-differential equations of neutral type as special cases. The adaptation of linear multistep methods, one-leg methods, and Runge-Kutta methods is considered. The emphasis is on the linear stability of numerical methods. It is proved that A -stable methods can inherit the asymptotic stability of underlying linear systems. Some general results of stability on explicit and implicit methods are also given.

In the present work, we are concerned with the derivation of continuous Runge-Kutta-Nystrom methods for the numerical treatment of second-order ordinary differential equations with Nystrom methods for the numerical treatment of second-order ordinary differential equations with periodic solutions. Numerical methods used for solving such problems are better to have the characteristics of high phase-lag order. First we analyse the construction algorithm for a high phase-lag order scaled extension of an explicit Runge-Kutta-Nystrom method. Using this procedure, we manage to construct a phase-lag order 14 continuous extension of a popular nine stages 8(6) order ERKN pair. In the literature, only phase-lag order 12 continuous extension of nine stage 8(6) ERKN pairs can be found, so the proposed scaling method has the higher, until now, dispersion order. Numerical tests for the proposed methods are done over various test problems.

This paper is concerned with the numerical solution of delay differential equations (DDEs). We focus on the error behaviour of general linear methods for stiff DDEs. A restricted type of interpolation procedure is considered for general linear methods. D -convergence properties of general linear methods with this interpolation procedure are investigated. Some extant results are unified.

An explicit two-step method exact for the scalar test equation y′ = λy, Re(λ) < 0 is presented in this paper. It is exponentially fitted, L-stable (thus, A-stable), and of order 2. With a new set of vector computations, we also extend directly the method to systems of ordinary differential equations. The numerical experiments demonstrate that this explicit two-step method is suitable for stiff systems.

This paper is concerned with the discrete numerical solution of coupled partial differential mixed problems with non-Dirichlet coupled boundary value conditions. By application of a discrete separation of variables method, the proposed numerical solution of the problem turns out to be the exact solution of certain coupled partial difference system, appearing from the discretization of the continuous partial differential system. Our approach avoids the iterative solution of algebraic systems which appears when one uses simple discretization methods. Existence, stability and the construction of solutions are considered.

An optimization-based domain decomposition method for the solution of partial differential equations is presented. The crux of the method is a constrained minimization problem for which the objective functional measures the jump in the dependent variables across the common boundaries between subdomains; the constraints are the partial differential equations. The existence of optimal solutions for the optimization problem is shown as is the convergence to the exact solution of the given problem. We then derive an optimality system of partial differential equations from which solutions of the domain decomposition problem may be determined. Finite element approximations to solutions of the optimality system are defined and analyzed as well as an eminently parallelizable gradient method for solving the optimality system. Then, the results of some numerical experiments and some concluding remarks are given. The latter includes the extension of the method to nonlinear problems such as the Navier-Stokes equations.