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In this article, we will consider the parallel implementation of the Yanenko algorithm for the two-dimensional heat equation, and the sweep method was used to numerically solve the heat equation. The implementation of the sequential program is carried out simply in two-part steps by the longitudinal-transverse run, however, parallelization of two fractional steps with an indefinite scheme is difficult due to the creation of inter-process communication of data. In the course of the study, a parallel data distribution with one-dimensional decompositions is shown in the application of the Yanenko method for calculating heat conductivity. The results of parallelization of this task using the 1D decomposition were obtained and acceleration and efficiency images were analyzed in order to evaluate the parallel program. Currently, modeling of processes by numerical solution of differential equations is widely used in every field of Science, the most common methods bring the differential problem to a system of linear algebraic equations, methods that solve such systems include various startup options. The emergence and development of computing systems using Multi-Core processors and graphics accelerators make the problem of startup parallelization relevant; the results of the study are used for teaching in research institutes and universities.

This paper deals with the qualitatively correct numerical solution of differential equations with known invariants. We use general linear methods with standard projection technique to preserve the invariants numerically by projecting the solution onto the desired manifold. The projection scheme admits the integration without any drift from the desired manifold and is computationally cost effective. The numerical experiments presented in this paper show the advantages of the proposed schemes.

Time-frequency analysis (TFA) for mechanical vibrations in non-stationary operations is the main subject of this article, concisely written to be an introducing tutorial comparing different time-frequency techniques for non-stationary signals. The theory was carefully exposed and complemented with sample applications on mechanical vibrations and nonlinear dynamics. A particular phenomenon that is also observed in non-stationary systems is the Sommerfeld effect, which occurs due to the interaction between a non-ideal energy source and a mechanical system. An application through TFA for the characterization of the Sommerfeld effect is presented. The techniques presented in this article are applied in synthetic and experimental signals of mechanical systems, but the techniques presented can also be used in the most diverse applications and also in the numerical solution of differential equations.

We present a novel motion planning strategy for the manipulation of elastic rods with two robotic arms. In previous work, it has been shown that the free configuration space of an elastic rod, i.e., the set of equilibrium shapes of the rod, is a smooth manifold of a finite dimension that can be parameterized by one chart. Thus, a sampling-based planning algorithm is straightforward to implement in the product space of the joint angles and the equilibrium configuration space of the elastic rod. Preliminary results show that planning directly in this product space is feasible. However, solving for the elastic rod's shape requires the numerical solution of differential equations, resulting in an excessive and impractical runtime. Hence, we propose to pre-compute a descriptor of the rod, i.e., a roadmap in the free configuration space of the rod that captures its main-connectivity. By doing so, we can plan the motion of any dual-arm robotic system over this roadmap with dramatically fewer solutions of the differential equations. Experiments using the Open Motion Planning Library (OMPL) show significant runtime reduction by an order of magnitude.

This work aims at evaluating the effects of magnetohydrodynamics (MHD) on the stagnation point flow along a nonlinear stretching/shrinking sheet in carbon nanotubes. Numerical methods for ordinary differential equations are obtained using the BVP4C solver in MATLAB software. Two kinds of base fluids, particularly water and kerosene, with single-walled and multi-walled carbon nanotubes are adopted in this analysis. The effect of various limitations on the Nusselt number and skin friction coefficient, as well as the temperature and velocity profiles are examined. From the numerical results, it is observed that non-unique solutions are visible for some limits of shrinking parameter. It is also found that nonlinear parameter and magnetohydrodynamic parameter act in widening the range of solution to exist. Therefore, the stability of flow was executed to identify the most stable solution between these two solutions. The stable and unstable flow of the first and second solutions, respectively, are confirmed.

A new phase-fitted Method for the numerical solution of Differential Equations in Chemistry is proposed in this paper.

The paper provides a brief overview of the possible scope of application of Lanchester-type models, and discusses some features of the numerical solution of differential equations of such models in stochastic setting. The interest in Lanchester models, which can be seen from publications in thematic periodicals, shows the effectiveness of such an apparatus in solving issues of rapid forecasting of changes in the number of opposing factions during hostilities. In addition, many modern systems of mathematical models of operations (combat operations) often use so-called express models. The main purpose of such models is to carry out rapid estimates of the dynamics of changes in the number of troops under different conditions of their use. The use of express models does not require a significant amount of source data and computing resource (unlike powerful model complexes). One possible direction of modifying Lanchester-type models to bring them closer to the realities of combat is to represent the equations of the model in stochastic form. Stochastic production allows to take into account random factors and elements of uncertainty, which influence the dynamics of changes in the number of opposing groups, and which to a certain extent are present in any combat. Unlike deterministic models, stochastic models require the use of special methods, the choice of a particular one can be justified by the requirements for their degree of convergence at the integration interval. The paper examines an example of using the Milstein method to numerically solve the differential equations of the Lanchester model in stochastic production. It is also shown that the convergence estimation of the method chosen for the numerical solution can serve to verify the correctness of its software implementation.

We present a procedure for finding feedback strategies numerically in a two-firm differential game (DG). To motivate the discussion, we start with the single-firm optimal control formulation of the...

The aim of this paper is to derive a new family of multi-block boundary value methods based on multi-block generalized Adams methods for numerical solution of stiff problems. Generally, boundary value methods easily circumvents the Dahlquist stability barrier in linear multistep formulas. This leads to a new stability framework for multi-block boundary value methods (MB $$_2$$ VMs) with the aid of the Wiener–Hopf factorization of a matrix polynomial, which generalizes the stability of initial multi-block methods in Chu and Hamilton (SIAM J Sci Stat Comput 8:342–535, 1987). The condition number of the arising Toeplitz matrix for the MB $$_2$$ VMs are very small, and in general are bounded. The MB $$_2$$ VMs are new classes of very large scale integration methods for the numerical solution of differential equations which output multi-block of solutions on application. Numerical experiments are presented and compared with some existing ones.

The article concerns methods of a structural curvilinear grid constructing in areas of geometrically complex shape and its evaluation from the quality point of view. Equidistribution methods based on differential equations were used to construct the grid at the boundary and inside the region. The numerical solution of differential equations was realized by the finite difference method. For the problems of uniform arrangement of grid nodes on the boundary and for the problems of constructing curved grids inside the region, implicit difference schemes were constructed and methods of scalar sweep and alternating directions were used. The results of numerical calculations are obtained and graphs of curved grids are presented for different numbers of grid nodes. The quality of the grid was studied according to four criteria such as orthogonality, elongation, convexity and adaptability, which corresponds to the division of the considered area into equal subdomains, i.e. cells.

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an inferential perspective, most notably the absence of explicit conditioning formula. This paper extends earlier work on linear PDEs to a general class of initial value problems specified by nonlinear PDEs, motivated by problems for which evaluations of the right-hand-side, initial conditions, or boundary conditions of the PDE have a high computational cost. The proposed method can be viewed as exact Bayesian inference under an approximate likelihood, which is based on discretisation of the nonlinear differential operator. Proof-of-concept experimental results demonstrate that meaningful probabilistic uncertainty quantification for the unknown solution of the PDE can be performed, while controlling the number of times the right-hand-side, initial and boundary conditions are evaluated. A suitable prior model for the solution of PDEs is identified using novel theoretical analysis of the sample path properties of Matérn processes, which may be of independent interest.

Two kinds of numerical solution of differential equations were first introduced to solve the problem of wind-induced vibration of bridge structures. Fluid-structure interaction (FSI) numerical simulation of a streamlined bridge section was realized using ANSYS Fluent as the computing platform and embedding user-defined function (UDF) in Fluent. Theoretical analysis indicated that the displacement gained by the FSI calculation model, where the grids near the bridge section were driven by conventional Newmark-beta (β) program embedded in UDF, differs from that of grid motion updated in Fluent. A corrected velocity method (i.e., CV Newmark-β) was proposed in this work to eliminate such an error. The corresponding FSI simulation model was then set up through the abovementioned method. For a concrete bridge section, the FSI simulation under low and high wind speeds was obtained by the models set up using conventional Newmark-β and CV Newmark-β programs. Result showed that the displacement time history curves obtained using different calculation models are consistent while the bridge section takes a small amplitude motion. The error from the displacement of conventional Newmark-β program and actual grid motion is small. In addition, the error of the displacement mismatch effect between this algorithm and the real motion of the grid can be ignored under low wind speed. While the bridge section takes a large amplitude flutter motion, the disparities of the displacement time history curves obtained using different calculation models are large. The error of the displacement mismatch effect between the conventional Newmark-β algorithm and the real motion of the grid must be considered under high wind speed. The displacement gained using the FSI calculation model set by the CV Newmark-β program is consistent with the real motion displacement of the grid motion updated in the Fluent program whether under low or high wind speed. Therefore, such a mismatch effect must be considered to establish a reasonable FSI calculation model while taking the numerical simulation of bridge flutter.

A method that employs a dual mesh, one for primary variables and another for dual variables, for the numerical solution of differential equations is presented. The formulation makes use of the trad...

Purpose. Investigation of the stress-strain state of the rotor of turbogenerator, which has damage in the form of cracks in the array of rotor barrel. Methodology. In the article mechanical stresses and deformations in the turbogenerator rotor's steel in a three-dimensional formulation are investigated by methods of mathematical modeling. Results. It is determined that the most mechanically region tensed is around the inner surface of the central opening of the turbogenerator rotor. In the nominal mode of operation, the crack in the central opening area of about 10 mm is critical for the turbogenerator. The appearance of a crack in the area of a small tooth contributes to breaking the tooth, which can damage the entire turbogenerator. Originality. The article continues analytical research and uses field methods based on the numerical solution of differential equations of structural mechanics using the finite element method. Practical value. The obtained results can be taken into account at the design stage of the turbogenerator, as well as when developing a system of diagnostic features for the diagnosis of the rotor damages. It is recommended in the process of turbogenerator rotor defectoscopy primarily to investigate the surface of the central hole of the rotor, because there is a high probability of cracks.

Geometric numerical integration concerns the development, analysis, and use of algorithms for the numerical solution of differential equations that preserve a geometric or qualitative property of t...

In this paper, we present new numerical methods to solve ordinary differential equations in both linear and nonlinear cases. we apply Daftardar-Gejji technique on theta-method to derive anew family of numerical method. It is shown that the method may be formulated in an equivalent way as a RungeKutta method. The stability of the methods is analyzed.

In this work, we construct and derive a new class of exponentially fitted two-derivative diagonally implicit Runge–Kutta (EFTDDIRK) methods for the numerical solution of differential equations with oscillatory solutions. First, a general format of so-called modified two-derivative diagonally implicit Runge–Kutta methods (TDDIRK) is proposed. Their order conditions up to order six are derived by introducing a set of bi-colored rooted trees and deriving new elementary weights. Next, we build exponential fitting conditions in order for these modified TDDIRK methods to treat oscillatory solutions, leading to EFTDDIRK methods. In particular, a family of 2-stage fourth-order, a fifth-order, and a 3-stage sixth-order EFTDDIRK schemes are derived. These can be considered as superconvergent methods. The stability and phase-lag analysis of the new methods are also investigated, leading to optimized fourth-order schemes, which turn out to be much more accurate and efficient than their non-optimized versions. Finally, we carry out numerical experiments on some oscillatory test problems. Our numerical results clearly demonstrate the accuracy and efficiency of the newly derived methods when compared with existing trigonometically/exponentially fitted implicit Runge–Kutta methods and two-derivative Runge–Kutta methods of the same order in the literature.

In computer studies of dynamic problems, numerical methods for solving differential equations are usually used. Essential importance in numerical calculations is guaranteed accuracy of the calculated solution, which depends on the accuracy of the computer used and the influence on the decision of inevitable errors of input data and rounding errors. Although the computational rules are built on the basis of the conditions for ensuring their possible growth with respect to the error, however, with a large number of steps, the deviation of the solution obtained by a numerical method from the exact one can be quite significant. Obtaining satisfactory estimates of the operational numerical solution of differential equations is a rather complicated task, which in many practical cases cannot be solved. Thus, an urgent task from a computational point of view is the development of approaches and methods that allow control of the computational process. In this paper, we consider the possibility of controlling the error of numerical solution by using the methods of parametric identification, which are widely used in solving practical problems of identifying linear and nonlinear systems. At the same time, the accuracy of the control should not depend on the reasons causing the error of the decision. The control process itself consists of the following steps: the parameters of the equations for which the resulting numerical solution is accurate are restored with some accuracy. The estimated parameters (the coefficients are compared with the coefficients of the original equations; the difference of the coefficients is the information that is used to evaluate the behavior of the solution on the restoration site (the recovery section is the segment of the numerical solution that is used for parametric identification).

In this chapter, from the point of view of Geometric Integration, i.e. the numerical solution of differential equations using integrators that preserve as many as possible the geometric/physical properties of them, we first introduce the concept of oscillation preservation for Runge–Kutta–Nystrom (RKN)-type methods and then analyse the oscillation-preserving behaviour of RKN-type methods in detail. This chapter is also accompanied by numerical experiments which show the importance of the oscillation-preserving property for a numerical method, and the remarkable superiority of oscillation-preserving integrators for solving nonlinear multi-frequency highly oscillatory systems.

In this work, we study hyperbolic conservation law in one space dimension with δ-singularities as the initial data. We use finite volume methods to find the sizes of pollution region. Firstly, we study finite volume method (FVM) with linear weights and weighted essentially non-oscillatory (WENO) scheme and apply both methods to linear partial differential equations without singularities to check the accuracy. Then we use both methods to find the numerical solutions and compute errors of linear equations with δ-singularities. Lastly, we use such results to find the size of pollution region of each method. These results show that the size of the pollution region is approximately of the order O(Δx), where Δx is the spatial mesh size.

The use of live demonstrations in courses related to numerical problem solving combined with PowerPoint presentations is advocated in order to stimulate student interest and involvement and enhance learning. The proposed approach is demonstrated by solving a system of ordinary differential equations as a boundary value problem, utilizing finite difference approximations for solving partial differential equations, and investigating ill-conditioning in multiple linear regression. The educational benefits of this approach are discussed in detail. © 2009 Wiley Periodicals, Inc. Comput Appl Eng Educ 17: 285–294, 2009; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae.20214

Spline collocation methods have evolved as valuable techniques for the solution of a broad class of problems covering ordinary and partial differential equations, functional equations, integral equations and integro-differential equations. The suboptimality of nodal spline collocation led researchers to seek spline collocation methods of optimal order. In their fundamental work, C. de Boor and B. Swartz showed that optimal rates of convergence can be attained by collocating at certain Gauss points in each subinterval of the partition on which the spline space is defined. The advantage of the methods over finite element Galerkin methods is that the calculation of the coefficients in the equations determining the approximate solution is very fast since no integrals need be evaluated or approximated. Basically, the method involves the determination of an approximate solution in a suitable set of functions, sometimes called trial functions, by requiring the approximate solution to satisfy the boundary conditions and the differential equation at certain points, called the collocation points.

A new application of the Bernstein polynomials based multiwavelets approach for the numerical solution of differential equations is given. In the proposed method, Bernstein polynomial multiwavelets are obtained by using orthonormality of Bernstein polynomials. We present the operational matrix of integration of Bernstein polynomial based multiwavelets basis which diminishes the taken differential equation into the system of algebraic equations for less demanding calculations. High accuracy of these results even in the case of a small number of polynomials is observed. The convergence and exactness are described by comparing the ascertained approximated solution and the known analytical solution. The error estimates of the approximate solution are given and also some comparative examples with figures are given to confirm the reliability and accuracy of the proposed method. Some physical problems that lead to the differential equations are examined by the proposed method.

The article deals with issues of increasing the accuracy of horizontal boring operation adjusted for the variable rigidity of the process system. Regularities of machining errors generation are established by mathematical modeling of operation. Mathematical model has been developed, which describes machining errors generation depending on the cutting modes, structural parameters of a tool, and physical properties of the material machined. Fluctuations in a mechanical system are described by equations of motion that consider mass displacements along two coordinates and a parametric change in the rigidity of the boring mandrel, and the components of the exciting forces are described by harmonic functions. A procedure for the numerical solution of differential equations is proposed based on the finite differences method. A program has been developed allowing solving the system of differential equations using the numerical method. A method has been developed for constructing a region for selecting rational modes: supply and revolutions at a pre-set cutting depth.

For the realization of integrated optical interconnect systems, optical waveguide components are needed. A promising approach to manufacturing graded-index waveguide components is their embedding in thin glass sheets by ion-exchange processes. Conventionally, the modeling of integrated waveguide components manufactured by thermal diffusion processes involves the numerical solution of differential equations. In this work, a semi-analytical approach for the modeling of the concentration profile is extended by the Gauss–Newton optimization algorithm. This is done with the specific purpose of accurately modeling single-mode input and output waveguide structures of multimode-interference-based (MMI-based) optical splitters. The constraints imposed by the multivariate manufacturing process for the realization of these MMI-based components are elaborated. Corresponding correlations between these constraints and the according modeling of the devices are established.

An algorithm is suggested for numerical solution of differential equations for velocity and pressure on a staggered grid. The algorithm ensures unconditional convergence of iterations for a correction to pressure and without it.

This study introduces new time-stepping strategies with built-in global error estimators. The new methods propagate the defect along with the numerical solution much like solving for the correction or Zadunaisky’s procedure; however, the proposed approach allows for overlapped internal computations and, therefore, represents a generalization of the classical numerical schemes for solving differential equations with global error estimation. The resulting algorithms can be effectively represented as general linear methods. Several explicit self-starting schemes akin to Runge–Kutta methods with global error estimation are introduced, and the theoretical considerations are illustrated in several examples.

In this paper we present a new class of direct numerical integrators of hybrid type for special third order ordinary differential equations (ODEs), y ′ ′ ′ = f ( x , y ) ; namely, hybrid methods for solving third order ODEs directly (HMTD). Using the theory of B-series, order of convergence of the HMTD methods is investigated. The main result of the paper is a theorem that generates algebraic order conditions of the methods that are analogous to those of two-step hybrid method. A three-stage explicit HMTD is constructed. Results from numerical experiment suggest the superiority of the new method over several existing methods considered in the paper.

This paper deals with the convergence and stability of linear multistep methods for a class of linear impulsive delay differential equations. Numerical experiments show that the Simpson’s Rule and two-step BDF method are of order p = 0 when applied to impulsive delay differential equations. An improved linear multistep numerical process is proposed. Convergence and stability conditions of the numerical solutions are given in the paper. Numerical experiments are given in the end to illustrate the conclusion.

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.

There have appeared in the literature a lot of k -step block methods for solving initial-value problems. The methods consist in a set of k simultaneous multistep formulas over k non-overlapping intervals. A feature of block methods is that there is no need of other procedures to provide starting approximations, and thus the methods are self-starting (sharing this advantage of Runge–Kutta methods). All the formulas are usually obtained from a continuous approximation derived via interpolation and collocation at k + 1 points. Nevertheless, all the k -step block methods thus obtained may be considered as different formulations of one of them, which results to be the most efficient and simple formulation of all of them. The theoretical analysis and the numerical experiments presented support this claim.

In this paper, an extended fourth-order Runge–Kutta method is studied to approximate the solutions of first-order fuzzy differential equations using a generalized characterization theorem. In this method, new parameters are utilized in order to enhance the order of accuracy of the solutions using evaluations of both f and f′, instead of using the evaluations of f only. The proposed extended Runge–Kutta method and its error analysis, which guarantees pointwise convergence, are given in detail. Furthermore, the accuracy and efficiency of the proposed method are demonstrated in a series of numerical experiments.

We extend the application of multiwavelet-based Bernstein polynomials for the numerical solution of differential equations governing AC circuits (LCR and LC). The operational matrix of integration is obtained from the orthonormal Bernstein polynomial wavelet bases, which diminishes differential equations into the system of linear algebraic equations for easy computation. It appeared that fewer wavelet bases gave better results. The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution. The error function was calculated and illustrated graphically for the reliability and accuracy of the proposed method. The proposed method examined several physical issues that lead to differential equations. HIGHLIGHTS Differential equations governing AC circuits are converted into the system of linear algebraic equations using Bernstein polynomial multiwavelets operational matrix of integration for easy computation The convergence and exactness were examined by comparing the calculated numerical solution and the known analytical solution The error function is calculated and shown graphically GRAPHICAL ABSTRACT

In this paper, two classes of implicit–explicit (IMEX) multistep finite-element methods are constructed to solve the nonlinear delay convection-diffusion-reaction equations. This equations are discretized with finite-element method in space and IMEX multistep methods (including IMEX one-leg methods and IMEX linear multistep methods) in time. The resulting schemes are efficient because it can reduce the coupling degree, and avoid solving the large-scale systems of nonlinear algebraic equations. The L2-norm error estimates with handling the breaking point and stability results for the proposed methods are obtained. Numerical results demonstrate the efficiency and accuracy of the approaches.

In this paper, we use a good technique to construct a high-order numerical scheme for the impulsive fractional ordinary differential equations (IFODEs). This technique is based on the so-called block-by-block method, which is a common method for the integral equations. In our approach, the classical block-by-block method is improved so as to avoid the coupling of the unknown solutions at each block step with an exception in the first two steps between two adjacent pulse points. The convergence and stability analysis of the scheme are given. It proves that the numerical solution converges to the exact solution with order 3+q for 0

In this article, we investigate the reproducing kernel method and the difference schemes method for solving the second-order partial differential equations. Numerical results have been shown to prove the efficiency of the methods. Results prove that the methods are very effective.

Explicit Runge–Kutta methods are standard tools in the numerical solution of ordinary differential equations (ODEs). Applying the method of lines to partial differential equations, spatial semidiscretisations result in large systems of ODEs that are solved subsequently. However, stability investigations of high-order methods for transport equations are often conducted only in the semidiscrete setting. Here, strong-stability of semidiscretisations for linear transport equations, resulting in ODEs with semibounded operators, are investigated. For the first time, it is proved that the fourth-order, ten-stage SSP method of Ketcheson (SIAM J Sci Comput 30(4):2113–2136, 2008) is strongly stable for general semibounded operators. Additionally, insights into fourth-order methods with fewer stages are presented.

In this article, one-step semi-Lagrangian method is investigated for computing the numerical solutions of the first-order time-dependent nonlinear partial differential equations in 1D with initial and boundary conditions. This method is based on Lagrangian trajectory or the integration from the departure points to the arrival points (regular nodes) and Runge–Kutta method for ordinary differential equations. The departure points are traced back from the arrival points along the trajectory of the path. The convergence and stability are studied for the implicit and explicit methods. The numerical examples show that those methods work very efficient for the time-dependent nonlinear partial differential equations.

We propose the multiplicative explicit Euler, multiplicative implicit Euler, and multiplicative Crank-Nicolson algorithms for the numerical solutions of the multiplicative partial differential equation. We also consider the truncation error estimation for the numerical methods. The stability of the algorithms is analyzed by using the matrix form. The result reveals that the proposed numerical methods are effective and convenient.

We introduce generalised finite difference methods for solving fully nonlinear elliptic partial differential equations. Methods are based on piecewise Cartesian meshes augmented by additional points along the boundary. This allows for adaptive meshes and complicated geometries, while still ensuring consistency, monotonicity, and convergence. We describe an algorithm for efficiently computing the non-traditional finite difference stencils. We also present a strategy for computing formally higher-order convergent methods. Computational examples demonstrate the efficiency, accuracy, and flexibility of the methods.

Motivated by the idea that staggered-grid methods give a greater stability and give energy conservation, this article presents a new family of high-order implicit staggered-grid finite difference methods with any order of accuracy to approximate partial differential equations involving second-order derivatives. In particular, we numerically analyze our new methods for the solution of the one-dimensional acoustic wave equation. The implicit formulation is based on the plane wave theory and the Taylor series expansion and only involves the solution of tridiagonal matrix equations resulting in an attractive method with higher order of accuracy but nearly the same computation cost as those of explicit formulation. The order of accuracy of the proposal staggered formulas are similar to the methods with conventional grids for a ( 2 M + 2 ) -point operator: the explicit formula is ( 2 M ) th-order and the implicit formula is ( 2 M + 2 ) th-order; however, the results demonstrate that new staggered methods are superior in terms of stability properties to the classical methods in the context of solving wave equations.

In this study we propose a numerical method for linear and nonlinear neutral pantograph equations. Most of the proposed numerical methods for this kind of equations was mainly introduced only for linear equations, but in the present paper we generalize our method for nonlinear cases, too. We prove that the numerical solution is convergent and some numerical experiment illustrate it.

The (asymptotic) behaviour of the second moment of solutions to stochastic differential equations is treated in mean-square stability analysis. This property is discussed for approximations of infinite-dimensional stochastic differential equations and necessary and sufficient conditions ensuring mean-square stability are given. They are applied to typical discretization schemes such as combinations of spectral Galerkin, finite element, Euler-Maruyama, Milstein, Crank-Nicolson, and forward and backward Euler methods. Furthermore, results on the relation to stability properties of corresponding analytical solutions are provided. Simulations of the stochastic heat equation illustrate the theory., Comment: 22 pages, 4 figures; deleted a section; shortened the presentation of results; corrected typos

The present work considers the numerical solution of differential equations that are obtained by space discretization (method of lines) of parabolic evolution equations. Main emphasis is put on the presence of mixed derivatives in the elliptic operator. An extension of the alternating-direction-implicit (ADI) approach to this situation is presented. Our stability analysis is based on a scalar test equation that is relevant to the considered class of problems. The novel treatment of mixed derivatives is implemented in third-order W-methods. Numerical experiments and comparisons with standard methods show the efficiency of the new approach. An extension of our treatment of mixed derivatives to 3D and higher dimensional problems is outlined at the end of the article.

In this paper, we propose efficient and reliable numerical methods to solve two notable non-integer-order partial differential equations. The proposed algorithm adapts the Fourier spectral method in space, coupled with the exponential integrator scheme in time. As an advantage over existing methods, our method yields a full diagonal representation of the non-integer fractional operator, with better accuracy over a finite difference scheme. We realize in this work that evolution equations formulated in the form of fractional-in-space reaction-diffusion systems can result in some amazing examples of pattern formation. Numerical experiments are performed in two and three space dimensions to justify the theoretical results. Simulation results revealed that pattern formation in a fractional medium is practically the same as in classical reaction-diffusion scenarios.

In this paper, we study efficient numerical integrators for linear and nonlinear systems of highly oscillatory second-order ordinary differential equations. The systems are reformulated as a first-order system, which is then transformed to adiabatic variables. The solution of the transformed system is a smoother function which is more accessible to numerical approximation than the original system. We develop Filon-type methods for linear systems by approximating the integral as a linear combination of function values and derivatives. We then present a special combination of Filon-type methods and waveform relaxation methods for nonlinear systems. Both types of methods can be used with far larger step sizes than those required by traditional schemes and their performance drastically improves as frequency grows, as are illustrated by numerical experiments.