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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.

Many phenomena in almost all areas of natural and engineering science are modelled by nonlinear differential equations. However, most of the explicit methods for time integration of nonlinear models fail to preserve some qualitative properties such as positivity of solutions. The major purpose of this study is to suggest a new explicit positivity preserving numerical method based on the exponential integrators. It is shown that the proposed method preserves the positivity of exact solution. Several examples are illustrated to confirm the theoretical result.

This paper is concerned with a numerical method for solving generalized fractional differential equation of Caputo–Katugampola derivative. A corresponding discretization technique is proposed. Numerical solutions are obtained and convergence of numerical formulae is discussed. The convergence speed arrives at O ( Δ T 1 − α ) . Numerical examples are given to test the accuracy.

This paper employs a differential quadrature scheme that can be used for solving linear and nonlinear partial differential equations in higher dimensions. Differential quadrature method with modified cubic B-spline basis functions is implemented to solve three-dimensional hyperbolic equations. B-spline functions are employed to discretize the space variable and their derivatives. The weighting coefficients are obtained by semi-explicit algorithm. The partial differential equation results into a system of first-order ordinary differential equations (ODEs). The obtained system of ODEs has been solved by employing a fourth stage Runge–Kutta method. Efficiency and reliability of the method has been established with five linear test problems and one nonlinear test problem. Obtained numerical solutions are found to be better as compared to those available in the literature. Simple implementation, less complexity and computational inexpensiveness are some of the main advantages of the scheme. Further, the scheme gives approximations not only at the knots but also at all the interior points in the domain under consideration. The scheme is found to be providing convergent solutions and handles different cases. High order time discretization using SSP–RK methods guarantee stability with respect to a given norm and a proper constraint on time step. Matrix method has been used for stability analysis in space and it is found to be unconditionally stable. The scheme can be used effectively to handle higher dimensional PDEs.

In this article, a numerical method is developed to obtain approximate solutions for a certain class of fractional differential equations. The method reduces the underlying differential equation to system of algebraic equations. An algorithm is presented to compute the coefficient matrix for the resulting algebraic system. Several examples with numerical simulations are provided to illustrate effectiveness of the method.

In this paper, a new splitting method, called canonical Euler splitting method (CES), is constructed and studied, which can be used for the efficient numerical solution of general nonlinear composite stiff problems in evolution equations of various type, such as ordinary differential equations (ODEs),źsemi-discrete unsteady partial differential equations (PDEs) and ordinary or partial Volterra functional differential equations (VFDEs),źand can significantly improve the computing speed on the basis of ensuring the computing quality. Stability, consistency and convergence theories of this method are established. A series of numerical experiments are given which check the efficiency of CES method and confirm our theoretical results.

This paper describes a numerical verification of solutions for infinite-dimensional functional equations based on residual forms and Newton-like iteration. The method is based upon a verification method previously developed by the authors. Several computer-assisted proofs for differential equations, including nonlinear partial differential equations, are presented.

The main purpose of this paper is to provide an efficient numerical approach for multi-order fractional differential equations based on a Tau-Collocation method. To do this, multi-order fractional differential equations transformed into a system of nonlinear algebraic equations in matrix form. Thus, by solving this system unknown coefficients are obtained. The fractional derivatives are described in the Caputo sense. The rate of convergence for the proposed method is established in the L w p norm. Some numerical example is also provided to illustrate our results. The results reveal that the method is very effective and simple.

For uncertain differential equations, we cannot always obtain their analytic solutions. Early researchers have described the Euler method and Runge-Kutta method for solving uncertain differential equations. This paper proposes a new numerical method-Adams method to solve uncertain differential equations. Some numerical experiments are given to illustrate the efficiency of our numerical method. Moreover, this paper also gives two numerical methods for calculating the extreme value and the time integral of solutions of uncertain differential equations.

In this paper, a numerical power series algorithm which is based on the improved Taylor matrix method is introduced for the approximate solution of Abel-type differential equations and also, Riccati differential equations. The technique is defined and illustrated with some numerical examples. The obtained results reveal that the method is very effective, simple and valid high accuracy. The method can be easily extended to other nonlinear equations.

In this manuscript, we implement a relatively new analytic iterative technique for solving time-space-fractional linear partial differential equations subject to given constraints conditions based on the generalized Taylor series formula. The solution methodology is based on generating the multiple fractional power series expansion solution in the form of a rapidly convergent series with minimum size of calculations. This method can be used as an alternative to obtain analytic solutions of different types of fractional linear partial differential equations applied in mathematics, physics, and engineering. Some numerical test applications were analyzed to illustrate the procedure and to confirm the performance of the proposed method in order to show its potentiality, generality, and accuracy for solving such equations with different constraints conditions. Numerical results coupled with graphical representations explicitly reveal the complete reliability and efficiency of the suggested algorithm.

We suggest a new constructive approach for the solvability analysis and approximate solution of general non-local boundary value problems for non-linear systems of ordinary differential equations with locally Lipschitzian non-linearities. The practical application of the techniques is explained on a numerical example.

In this paper, three types of third-order partial differential equations (PDEs) are classified to be third-order PDE of type I, II and III. These classes of third-order PDEs usually occur in many subfields of physics and engineering, for example, PDE of type I occurs in the impulsive motion of a flat plate. An efficient numerical method is proposed for PDE of type I. The PDE of type I is converted to a system of third-order ordinary differential equations (ODEs) using the method of lines. The system of ODEs is then solved using direct Runge-Kutta which we derived purposely for solving special third-order ODEs of the form y ? = f ( x , y ) . Simulation results showed that the proposed RKD-based method is more accurate than the existing finite difference method.

In this paper, we study an application of the moving mesh method to the porous medium equation with absorption and variable exponents of nonlinearity in 2D domains with moving boundaries. The boundary’s movement is governed by an equation prompted by the Darcy law and the spatial discretization is defined by a triangulation of the domain. At each finite element, the solution is approximated by piecewise polynomial functions of degree r ⩾ 1 using Lagrange interpolating polynomials in area coordinates. The vertices of the triangles move according to a system of differential equations which is added to the equations of the problem. The resulting system is converted into a system of ordinary differential equations in time variable, which is solved using a suitable integrator. The integrals that arise in the system of ordinary differential equations are calculated using the Gaussian quadrature. Finally, we present some numerical results of application of this technique.

An eigenfunction approach to compute the particular solution of constant coefficient ordinary differential equations is presented. It is shown that the exponentials are the eigenfunctions of such equations. Solutions corresponding to products of powers and exponentials are obtained. The singular case is studied and a fast algorithm for its implementation is presented.

In this note, the problem on the exponential stability in mean square moment of mild solution to impulsive neutral stochastic partial functional differential equations is considered by employing the inequality technique. Some sufficient conditions are established for the concerned problem, and some existing results are generalized and improved. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained result.

In this paper, we present a new approach to solve nonlocal initial-boundary value problems for linear and nonlinear parabolic and hyperbolic partial differential equations subject to initial and nonlocal boundary conditions of integral type. We first transform the given nonlocal initial-boundary value problems of integral type for the linear and nonlinear parabolic and hyperbolic partial differential equations into local Dirichlet initial-boundary value problems, and then use a relatively new modified Adomian decomposition method (ADM). Furthermore we investigate the Fourier-Adomian method, which also does not require any a priori assumptions on the solution, for the solution of nonlocal initial-boundary value problems combined with our new approach. Several examples are presented to demonstrate the efficiency of the ADM.

Analytical properties of a nonlinear singular second order boundary value problem in ordinary differential equations posed on an unbounded domain for the density profile of the formation of microscopic bubbles in a nonhomogeneous fluid are discussed. Especially, sufficient conditions for the existence and uniqueness of solutions are derived. Two approximation methods are presented for the numerical solution of the problem, one of them utilizes the open domain Matlab code bvpsuite. The results of numerical simulations are presented and discussed.

In this paper, a new matrix approach for solving second order linear partial differential equations (PDEs) under given initial conditions has been proposed. The basic idea includes integrating from the considered PDEs and transforming them to the associated integro-differential equations with partial derivatives. Therefore, Bernoulli operational matrices of differentiation and integration together with the completeness of Bernoulli polynomials can be used for transforming integro-differential equations to the corresponding algebraic equations. A rigorous error analysis in the infinity norm is given provided that the known functions and the exact solution are sufficiently smooth and bounded. A numerical example is included to demonstrate the validity and the applicability of the technique. The results confirm the theoretical prediction.

This article presents numerical methods for solving second-order ordinary differential equations. These methods are based on Hermite polynomials, which makes them more computationally effective than, for example, the classical fourth-order Runge-Kutta method. In addition, the presented algorithms were modified to reduce the CPU time required. Hermite polynomials are not very sensitive to the Runge phenomenon; moreover, the numerical errors of interpolation are relatively small for large time steps, which is an advantage. These methods are presented in the form of pseudo-code for easier application. The presented approach to numerical methods is a result of simulated, strongly non-linear vibrations with contact phenomena such as Coulomb friction and impact.

We discuss the least squares methods for solving partial differential equations on arbitrary polygon domain by using Bezier control points. It uses triangular Bezier patches of degree n with C k continuity to approximate the exact solution of partial differential equations. Numerical examples are provided to illustrate the proposed method is flexible.

Radially symmetric solutions of many important systems of partial differential equations can be reduced to systems of special ordinary differential equations. A numerical solver for initial value problems for such systems is developed based on Matlab, and numerical bifurcation diagrams are obtained according to the behavior of the solutions. Various bifurcation diagrams of coupled Schrodinger equations from nonlinear physics are obtained, which suggests the uniqueness of the ground state solution.

In this article, we propose a two-level compact implicit off-step discretization for the solution of the system of 2D quasi-linear parabolic partial differential equations subject to suitable initial and boundary conditions. New methods for the estimates of first order space derivatives of the solution are also derived. These methods are fourth order accurate in space and second order accurate in time, and involve only nine spatial grid points of a single compact cell. We further develop the alternating direction implicit (ADI) scheme for a general linear parabolic equation which is shown to be unconditionally stable for the heat equation in polar coordinates. The proposed methods are directly applicable to singular problems without the need of any special technique, which is the main advantage of this work. The method is effectively applied to the time dependent Navier–Stokes’ model equations in polar coordinates. Numerical examples are provided to illustrate the accuracy of the methods.

In this paper we study the numerical solution of initial-boundary problem for a class of partial integro-differential equations. The quasi wavelet method is proposed to handle the spatial derivatives while the forward Euler method is used to discretize the temporal derivatives. Detailed discrete schemes are given and some numerical experiments are included to demonstrate the effectiveness of the discrete technique. The comparisons of the present numerical results with the exact analytical solutions show that the quasi wavelet based numerical method has distinctive local property and can achieve accurate results.

Linear Volterra integro-differential equations with infinite delay are studied. Sufficient conditions for exponential asymptotic stability of linear time-varying equations with delay are given. Then several explicit criteria for exponential asymptotic stability of linear time-invariant equations are presented. Two examples are given to illustrate the obtained results.

This paper is concerned with the uniform exponential stability of ordinary and delay dynamic equations. After revealing the equivalence between various types of uniform exponential stability definitions on time scales with bounded graininess, and demonstrating their relation when the graininess is arbitrary, we confine our attention to the uniform exponential stability of ordinary dynamic equations. We introduce and prove the Bohl-Perron criterion for delay dynamic equations: if for any bounded right-hand side, the solution of the delay dynamic equation with bounded coefficients and delays is bounded, then the trivial solution of the equation is uniformly exponentially stable. We also obtain some corollaries of this criterion. Based on these results, explicit exponential stability tests are derived for delay dynamic equations with nonnegative coefficients, which are illustrated with an example on a nonstandard time scale. (C) 2012 Elsevier Inc. All rights reserved.

New types of chaotic attractors for some 3-D autonomous systems of ordinary quadratic differential equations are founded. Examples are given.

Initial boundary value problems of Dirichlet type for first order partial functional differential equations are considered. Explicit difference schemes and implicit difference methods are investigated. Sufficient conditions for the convergence of approximate solutions are given and comparisons of the methods are presented. It is proved that assumptions on the regularity of given functions are the same for both the methods. It is shown that conditions on the mesh for explicit difference schemes are more restrictive than suitable assumptions for implicit methods. There are implicit difference schemes which are convergent and corresponding explicit difference methods are not convergent. Error estimates for both the methods are constructed. Numerical examples are presented.

In this paper, we describe a numerical method based on fitted operator finite difference scheme for the boundary value problems for singularly perturbed delay differential equations with turning point and mixed shifts. Similar boundary value problems are encountered while simulating several real life processes for instance, first exit time problem in the modelling of neuronal variability. A rigorous analysis is carried out to obtain priori estimates on the solution and its derivatives for the considered problem. In the development of numerical methods for constructing an approximation to the solution of the problem, a special type of mesh is generated to tackle the delay term along with the turning point. Then, to develop robust numerical scheme and deal with the singularity because of the small parameter multiplying the highest order derivative term, an exponential fitting parameter is used. Several numerical examples are presented to support the theory developed in the paper.

Third order nonlinear ordinary differential equations, subject to appropriate boundary conditions arising in fluid dynamics, are solved using three different methods viz., the Dirichlet series, method of stretching of variables, and asymptotic function method. Similarity transformations are used to convert the governing partial differential equations into nonlinear ordinary differential equations. The numerical results obtained from the above methods for various problems are given in terms of skin friction. Our study revealed that the results obtained from these methods agree well with those of direct numerical simulation of ordinary differential equations. Also, these methods have advantages over pure numerical methods in obtaining derived quantities such as velocity profile accurately for various values of the parameters at a stretch.

This paper discussed how to solve the polynomial ordinary differential equations. At first, we construct the theory of the linear equations about the unknown one variable functions with constant coefficients. Secondly, we use this theory to convert the polynomial ordinary differential equations into the simultaneous first order linear ordinary differential equations with constant coefficients and quadratic equations. Thirdly, we work out the general solution of the polynomial ordinary differential equations which is no longer concerned with the differential. Finally, we discuss the necessary and sufficient condition of the existence of the solution.

Two autonomous, nonlinear, third-order ordinary differential equations whose dynamics can be represented by second-order nonlinear ordinary differential equations for the first-order derivative of the solution are studied analytically and numerically. The analytical study includes both the obtention of closed-form solutions and the use of an artificial parameter method that provides approximations to both the solution and the frequency of oscillations. It is shown that both the analytical solution and the accuracy of the artificial parameter method depend greatly on the sign of the nonlinearities and the initial value of the first-order derivative.

We assess the performance of an exponential integrator for advancing stiff, semidiscrete formulations of the unsaturated Richards equation in time. The scheme is of second order and explicit in nature but requires the action of the matrix function φ ( A ) = A −1 ( e A − I ) on a suitability defined vector v at each time step. When the matrix A is large and sparse, φ ( A ) v can be approximated by Krylov subspace methods that require only matrix–vector products with A . We prove that despite the use of this approximation the scheme remains second order. Furthermore, we provide a practical variable-stepsize implementation of the integrator by deriving an estimate of the local error that requires only a single additional function evaluation. Numerical experiments performed on two-dimensional test problems demonstrate that this implementation outperforms second-order, variable-stepsize implementations of the backward differentiation formulae.

This paper presents a new method for validating existence and uniqueness of the solution of an initial value problems for fractional differential equations. An algorithm selecting a stepsize and computing a priori constant enclosure of the solution is proposed. Several illustrative examples, with linear and nonlinear fractional differential equations, are given to demonstrate the effectiveness of the method.

The motivation underlying this contribution has been to complete some of the topics concerning exact schemes for numerically solving ordinary differential equations. A procedure for obtaining differential systems exactly solved by a given finite-difference method is described. Examples illustrating the application of the procedure for obtaining first-order, second-order and systems of differential equations exactly solved by different numerical methods are given. Among the numerical methods considered there are the Trapezoidal Rule, the two-step Adams–Bashforth method and the Numerov method. Some numerical examples are presented to provide evidence that the procedure works properly.

A two-dimensional differential transformation method is employed to reduce the partial differential equations of the non-continuous thermal conductive boundary value problem to a Taylor series in a polynomial form. The partial differential equations are solved by the two-dimensional T-spectra method of differential transformation and by the use of trial initial polynomial conditions. The investigative parameters include the time-step of the differential transformation and the number of sub-domain segments. The numerical simulation results indicate that the proposed approach using the two-dimensional T-spectra method of differential transformation is applicable to the solution of non-continuous thermal conductive boundary value problems.

In this paper, we generalize the well-known Lyapunov-type inequality for second-order linear differential equations to a class of even-order linear differential equations, the results of this paper are new and generalize some known inequalities in the literatures.

This paper is concerned with the numerical solution to initial value problems of nonlinear delay differential equations of neutral type. We use A-stable linear multistep methods to compute the numerical solution. The asymptotic stability of the A-stable linear multistep methods when applied to the nonlinear delay differential equations of neutral type is investigated, and it is shown that the A-stable linear multistep methods with linear interpolation are GAS-stable. We validate our conclusions by numerical experiments.

Fuzzy differential equations (FDEs) generalize the concept of crisp initial value problems. In this article, we deal with the numerical solution of FDEs. The notion of convergence of a numerical method is defined and a category of problems which is more general than the one already found in the numerical analysis literature is solved. Efficient s-stage Runge–Kutta methods are used for the numerical solution of these problems and the convergence of the methods is proved. Several examples comparing these methods with the previously developed Euler method are displayed.

A method for solving some nonlinear fractional parabolic partial differential equations is considered. Using this method, new exact solutions are obtained. By introducing suitable transformation, we obtain a system of fractional differential equations. This system will generate the exact solutions. The efficiency of the considered method can be demonstrated for a large variety of equations more general than Burgers, Murrary and Fishers equations.

In the present paper a numerical technique is developed for the approximate solution of second-order partial differential equations (PDEs) with variable coefficients in three dimensions. With the temporary introduction of two unknown auxiliary functions of the coordinate system the initial equation is separated into three parts that are reduced to ordinary differential equations, one for each dimension, that are discretized with a finite difference scheme. The use of suitable manipulations and the elimination of the unknown auxiliary functions, gives finally a linear system of algebraic equations where the matrix of the coefficients of the unknowns is diagonally dominant, a prerequisite for the rapid convergence of the iterative procedure. The efficiency and accuracy of the proposed numerical scheme is validated by its application to two test problems of fluid mechanics which have exact solutions. The numerical results based on the present technique are more accurate than those obtained by either the standard relaxation treatment with central differences or the ADI method when the contribution of the first-derivative terms in the initial equation is dominant. In all cases the comparison of the numerical results with those of the analytical solution, demonstrates the reliability of the presented numerical code.

Most methods for the determination of approximate solutions of nonlinear ordinary differential equations require that the nonlinearities be sufficiently differentiable with respect to the dependent variable and its derivatives. In this paper, a non-iterative method for obtaining approximate solutions of nonlinear ordinary differential equations which does not require the derivatives of the nonlinearities is presented and its convergence is proved. The method is also generalized for the determination of the periodic solutions of autonomous, nonlinear ordinary differential equations by introducing as dependent variable the unknown frequency of oscillation. The method does not require the presence of small parameters and can be used in a piecewise fashion.

Spatial discretization of many evolutionary partial differential equations leads to stiff semilinear systems of ordinary differential equations. Exponential integrators solve exactly the linear part, which is assumed to be responsible of the stiffness of the problem, while the nonlinear terms are approximated numerically. In this paper, a variable step implementation of the exponential time differencing methods of multistep type is considered. A local error estimate of exponential type, whose computation is for free, is introduced and some implementation issues for diagonalizable matrices are addressed. Numerical results showing the behaviour of the proposed variable step implementation are provided too.

This paper deals with the stability analysis of the analytic and numerical solutions of linear impulsive differential equations. The numerical method with variable stepsize is defined, the conditions that the numerical solutions preserve the stability property of the analytic ones are obtained and some numerical experiments are given.

In this paper an initial value problem for a system of singularly perturbed ordinary differential equations is considered. A parameter robust computational method is constructed and it is proved that it gives essentially first order parameter-uniform convergence in the maximum norm. Numerical results are presented in support of the theory.

In this article, He’s variational iteration method (VIM) is employed to solve a system of differential equations of first order. Since every ordinary differential equations of higher order can be converted into a system of differential of the first order, this method can be used for solving most differential equations. Some examples are presented to show the ability of the method for linear and non-linear systems of differential equations. The results reveal that the method is very effective and simple.

Two families of Rothe’s methods that are based on the discretization of the time variable and keeping the spatial one continuous for the solution of second-order hyperbolic equations with damping and nonlinear source terms are presented. The first family is based on time integration and results in a Volterra integro-differential equation which, upon approximating certain time integrals, can be written as an ordinary differential equation in space. Upon discretizing the spatial derivatives by means of finite difference formula, this family results in nonlinear algebraic equations which contain exponential terms that depend on the time step, and its accuracy is shown to be lower than that of an explicit second-order accurate discretization in both space and time of second-order hyperbolic equations. The second family of methods is based on the discretization of the time derivatives and time linearization, and results in linear ordinary differential equations in space. Upon freezing the coefficients of these equations, one can integrate analytically the resulting linear ordinary differential equations to obtain piecewise exponential solutions which are continuous in space, and three-point finite difference equations which depend exponentially on the time step, spatial grid size, and the diffusion, relaxation, damping and reaction times. The finite difference equations are shown to result in non-diagonally dominant matrices unless the time step is smaller than the characteristic relaxation, diffusion, damping and reaction times. To avoid this problem, two Rothe’s techniques that do not account for the Jacobian of the reaction terms in the ordinary differential operator are developed, and it is shown that the Rothe’s techniques belonging to the second family are as accurate as the linearly implicit methods presented in Part I provided that the relaxation time is smaller than the critical time and the solution does not oscillate in space. When the relaxation time is greater than the critical one, Rothe’s methods have been found to be less accurate than the finite difference techniques of Part I. The Rothe’s techniques belonging to the second family are generalized to systems of nonlinear second-order hyperbolic equations and mixed systems of parabolic and second-order hyperbolic partial differential equations.

In the present paper, the dynamical differential equations with initial conditions is converted into the model of linear operator action, in which the linear operator is just the infinitesimal generator of the solver of the differential equations. And the resolvent of the linear operator is the Laplace transform of the solver of original differential equations. So the solver of original differential equations can be obtained by inversing the Laplace transform of its resolvent. An iterative algorithm for nonlinear differential equations with initial condition is easily presented by means of numerical Laplace transform inversion. The numerical examples show that the method of this paper is effective.

In present paper numerical schemes are developed for obtaining approximate solutions to the mixed problem for one-dimensional diffusion equation with variable space operator.The method of lines semidiscretization approach is used to transform the model partial differential equation into a system of first-order linear ordinary differential equations. The partial derivative with respect to the space variable is approximated by the first and second-order finite-difference approximation. For the solution of the resulting system of first-order ordinary differential equations we apply the first and second order of accuracy difference schemes. Stability estimates for the solution of these difference schemes are established. Numerical techniques are developed by applying a procedure of the solution of first order linear difference equation with matrix coefficients. The algorithms are tested on a model problem in biofluid mechanics. Two regions are considered close to the endothelial cells (EC) which can be modeled taking the mixed problem for one-dimensional diffusion equation with variable space operator.