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International audience; In this work, we focus on the numerical approximation of a hybrid fluid-kinetic plasma model for electrons, in which energetic electrons are described by a Vlasov kinetic model whereas a fluid model is used for the cold population of electrons. First, we study the validity of this hybrid modelling in a two dimensional context (one dimension in space and one dimension in velocity) against the full (stiff) Vlasov kinetic model and second, a four dimensional configuration is considered(one dimension in space and three dimensions in velocity) following [1]. To do so, we consider two numerical Eulerian methods. The first one is based on the Hamiltonian structure of the hybrid system and the second approach, which is based on exponential integrators, enables to derive high order integrator and remove the CFL condition induced by the linear part. The efficiency of these methods, which are combined with an adaptive time stepping strategy, are discussed in the different configurations and in the linear and nonlinear regimes.

Exponential integrators based on discrete gradient methods are applied to non-canonical Hamiltonian systems with added linear forcing/damping terms, which may be time-dependent. Changes in the dynamics, such as conservation of energy or Casimirs, which result from inclusion of the linear forcing/damping terms, are not exactly preserved by standard discrete gradient methods. However, those changes are shown to be exactly preserved by the exponential integrators in special circumstances. The methods are also symmetric, second order, and linearly stable. To demonstrate advantages in both accuracy and efficiency over other standard methods, the exponential integrators are applied to a three dimensional Lotka-Volterra system and a damped/driven Ablowitz-Ladik system.

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 paper, we present the Taylor polynomial solutions of system of higher order linear integro-differential Volterra-Fredholm equations (IDVFE). This method transforms IDVFE into the matrix equations which correspond to a system of linear algebraic equations. Some numerical results are also given to illustrate the efficiency of the method. Keywords: Taylor polynomial, Integro-differential equation, Volterra-Fredholm equations, Convergence analysis

We present a new symmetric linearly implicit exponential integrator that preserves the polynomial first integrals or the Lyapunov functions for the conservative and dissipative stiff equations, respectively. The method is tested by both oscillated ordinary differential equations and partial differential equations, e.g., an averaged system in wind-induced oscillation, the Fermi–Pasta–Ulam systems, and the polynomial pendulum oscillators. The numerical simulations confirm the conservative properties of the proposed method and demonstrate its good behavior in superior running speed when compared with fully implicit schemes for long-time simulations.

In this paper, we derive and analyze symmetric exponential integrators for charged-particle dynamics in a strong and constant magnetic field. We first present the scheme of exponential integrators and then establish the symmetry conditions for the methods. Explicit symmetric exponential integrators up to order four are constructed on the basis of the symmetry conditions. In order to show the remarkable performances of new symmetric methods in comparison with symmetric Runge–Kutta methods including the 2-stage Gauss method, two numerical experiments are carried out and the numerical results demonstrate the super numerical behavior.

PART I: ORDINARY DIFFERENTIAL EQUATIONS Advances in the Asymptotic and Numerical Solution of Linear Ordinary Differential Equations, F.W.J. Olver Some Unsolved Problems in Asymptotics, R. Wong Periodic Solutions and Heteroclinic Cycles in the Convection Model of a Rotating Fluid Layer, J. Li and X.H. Zhao The Equivalence of Exponential Stability for Impulsive Time-Delay Differential Systems, Z.-H. Guan, Y.-C. Zhou, and X.-P. He Conditions for Identity of Bifurcations in Cubic Hamiltonian Systems with Symmetry or Nonsymmetry Perturbations, Z. Liu, H. Cao, and J. Li PART II: PARTIAL DIFFERENTIAL EQUATIONS Long Time Behavior for the Generalized Ginzburg-Landau Equations, B. Guo The Inverse Scattering Transform for a Variable-Coefficient KdV Equations (with Applications to Shallow Water Waves), H.-H. Dai The Semigroup Theory and Abstract Linear Equations, G. Yang A Unified Approach Towards Nonlinear Parabolic Equations with Strong Reaction in Rn, Y.-W. Qi Global Existence of Smooth Solution to Boltzmann-Poisson System in Semiconductor Physics, G. Cui and Y. Wang Analytical Methods for a Selection of Elliptic Singular Perturbation Problems, N.M. Temme Exponential Attractors of the Strongly Damped Nonlinear Wave Equations, Z. Dai and B. Guo Generalized Isovorticity Principle for Ideal Magnetohydrodynamics, V.A. Vladimirov and K.I. Ilin Scroll Waves in Excitable Media and the Motion of Organization Center, Q. Lu and S. Liu Transport Equations for a General Class of Evolution Equations, M.Z. Guo and X.P. Wang a-Times Integrated Cosine Function, G. Yang Identifying Parameters in Elliptic Systems by Finite Element Methods with Multi-Level Initializing, Y.F. Seid and J. Zou Techniques Monotone Difference Schemes for Two Dimensional Nonhomogeneous Conservation Laws, T. Tang and Z.-H. Teng

The method of Riccati’s equation is applied to find a stability criterion for systems of two first-order linear ordinary differential equations. The obtained result is compared for a particular example with results obtained by the Lyapunov and Bogdanov methods, by using estimates of solutions of systems in terms of the Losinskii logarithmic norms, and by the freezing method.

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.

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.

In this article, it is shown that the combined use of exponential operators and integral transform provides a powerful tool to solve certain type of fractional PDEs. It is shown that exponential operators are powerful and effective method for solving certain differential equations and fractional Fokker-Planck equations with non-constant coefficients. Constructive examples are also provided.

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.

We present a numerical approach for the approximate solutions of first order initial value problems (IVP) by using unsupervised radial basis function networks. The proposed unsupervised method is able to solve IVPs with high accuracy. In order to demonstrate the efficiency of the proposed approach, we also compare its solutions with the solutions obtained by a previously proposed neural network method for representative examples.

This work is devoted to the numerical methods for solving the first-kind boundary value problem for a linear second-order differential equation with a deviating argument in minor terms. The sufficient conditions of the one-valued solvability are established, and the a priori estimate of the solution is obtained. For the numerical solution, the problem studied is reduced to the equivalent boundary value problem for an ordinary linear differential equation of fourth order, for which the finite-difference scheme of second-order approximation was built. The convergence of this scheme to the exact solution is shown under certain conditions of the solvability of the initial problem. To solve the finite-difference problem, the method of five-point marching of schemes is used.

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.

We consider Hamiltonian partial differential equations u tt +|∂ x u|+σu = f(u); x ∈ T; t ∈ ℝ; with periodic boundary conditions, where f(u) is a real-analytic function of the form f(u) = u5 + o(u5) near u = 0; σ ∈ (0; 1) is a fixed constant, and T = ℝ/2πℤ: A family of quasi-periodic solutions with 2-dimensional are constructed for the equation above with π ∈ (0; 1)ℚ: The proof is based on infinite-dimensional KAM theory and partial Birkhoff normal form.

We discuss the existence of solutions of initial value problems for a class of hybrid fractional neutral differential equations. To prove the main results, we use a hybrid fixed point theorem for the sum of three operators. We also derive the dependence of a solution on the initial data and present an example to illustrate the results.

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.

The purpose of this paper is to analyze a class of nonlinear fractional differential models generated by impulsive effects. Based on variational methods, some new results are obtained in regard to the existence of solutions. Moreover, our results enrich and extend some existing conclusions.

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.

Many numerical methods have been developed for nonlinear fractional integro-differential Volterra’s population model (FVPG). In these methods, to approximate a function on a particular interval, only a restricted number of points have been employed. In this research, we show that it is possible to use the fuzzy transform method (F-transform) to tackle with FVPG. It makes the F-transform preferable to other methods since it can make full use of all points on this interval. We also make a comparison showing that this method is less computational and is more convenient to be utilized for coping with nonlinear integro-differential equation (IDEs), fractional nonlinear integro-differential equation (FIDEs), and fractional ordinary differential equations (FODEs).

In this article, we consider the existence and uniqueness, and the asymptotic behavior of the strong solution for the following neutral stochastic partial functional differential equations. where A(t): V → V* is a linear bounded operator. , and are some appropriate measurable functions. By establishing the variational framework, the existence and uniqueness for the strong solution of such equations is shown under the coercivity condition. Then, the asymptotic behavior of the strong solution is investigated and some well-known results are extended. As a by-product, the exponential stability in mean square and almost sure exponential stability for a strong solution of neutral stochastic partial differential equations with delays are also discussed by utilizing the integral inequality. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained results.

We address the challenging problem of the exponential stability of nonlinear time-varying functional differential equations of neutral type. By a novel approach, we present explicit sufficient conditions for the exponential stability of nonlinear time-varying neutral functional differential equations. A discussion of the obtained results and illustrative examples are given.

WOS: 000448859600011, In this paper, we have obtained the numerical solutions of complex differential equations with variable coefficients by using the Legendre Polynomials and we have performed it on two test problems. Then, we applied with different technical of error analysis to the test problems. When we compared exact solutions and numerical solutions of tables and graphs, we realized that our method is reliable, practical and functional.

A class of generic spatially extended fractional reaction-diffusion systems that modelled predator-prey interactions is considered. The first order time derivative is replaced with the Caputo fractional derivative of order γ ∈ (0, 1). The local analysis where the equilibrium points and their stability behaviours are determined is based on the adoption of qualitative theory for dynamical systems ordinary differential equations. We derived conditions for Hopf bifurcation analytically. Most significantly, existence conditions for a unique stable limit cycle in the phase plane are determined analytically. Our analytical findings are in agreement with the numerical results presented in one and two dimensions. The system of fractional nonlinear reaction-diffusion equations has demonstrated the usefulness of understanding the dynamics of nonlinear phenomena.

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.

In this article, we propose a new two-level implicit method of accuracy two in time and three in space based on spline in compression approximations using two off-step points and a central point on a quasi-variable mesh for the numerical solution of the system of 1D quasi-linear parabolic partial differential equations. The new method is derived directly from the continuity condition of the first-order derivative of the spline function. The stability analysis for a model problem is discussed. The method is directly applicable to problems in polar systems. To demonstrate the strength and utility of the proposed method, we solve the generalized Burgers-Fisher equation, generalized Burgers-Huxley equation, coupled Burgers-equations and heat equation in polar coordinates. We demonstrate that the proposed method enables us to obtain high accurate solution for high Reynolds number.

In this paper, a thorough analysis is given for the order which is observed when integrating evolutionary linear partial differential equations with Lawson methods. The analysis is performed under the general framework of C0-semigroups in Banach spaces and hence it can be applied to the numerical time integration of many initial boundary value problems which are described by linear partial differential equations. Conditions of regularity and annihilation at the boundary of these problems are then stated to justify the precise order which is observed, including fractional order of convergence.

This article is concerned to the investigation of extremal solutions for a system of fractional order differential equations with coupled integral boundary value problem. In initial stage, we establish a comparison result and then using the iterative technique of monotone type together with the procedure of extremal solutions, we develop sufficient conditions to obtain the solutions for the considered fractional differential system. Moreover, the investigated results are also justified by providing suitable examples.

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.

Numerical methods are proposed for solving some problems for a system of linear ordinary differential equations in which the basic conditions (which are generally nonlocal ones specified by a Stieltjes integral) are supplemented with redundant (possibly nonlocal) conditions. The system of equations is considered on a finite or infinite interval. The problem of solving the inhomogeneous system of equations and a nonlinear eigenvalue problem are considered. Additionally, the special case of a self-adjoint eigenvalue problem for a Hamiltonian system is addressed. In the general case, these problems have no solutions. A principle for constructing an auxiliary system that replaces the original one and is normally consistent with all specified conditions is proposed. For each problem, a numerical method for solving the corresponding auxiliary problem is described. The method is numerically stable if so is the constructed auxiliary problem.

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.

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 article, a recently introduced iterative method based on Green's functions and fixed-point iteration schemes is presented for the approximate solutions of delay and functional differential ...

Based on the collocation technique, we introduced a unifying approach for deriving a family of multi-point numerical integrators with trigonometric coefficients for the numerical solution of periodic initial value problems. A practical 3-point numerical integrator was presented, whose coefficients are generalizations of classical linear multistep methods such that the coefficients are functions of an estimate of the angular frequency ω. The collocation technique yields a continuous method, from which the main and complementary methods are recovered and expressed as a block matrix finite difference formula that integrates a second-order differential equation over non-overlapping intervals without predictors. Some properties of the numerical integrator were investigated and presented. Numerical examples are given to illustrate the accuracy of the method.

In this paper, we consider a numerical scheme for a class of non-linear time delay fractional diffusion equations with distributed order in time. This study covers the unique solvability, convergence and stability of the resulted numerical solution by means of the discrete energy method. The derivation of a linearized difference scheme with convergence order O(τ+(Δα)4+h4) in L∞-norm is the main purpose of this study. Numerical experiments are carried out to support the obtained theoretical results. © 2016 Elsevier B.V.

We discuss (survey) some recent results on several aspects of complex analytic and meromorphic solutions of linear and nonlinear partial differential equations, with main attention given to those of the authors and their collaborators, and also give some new results on these equations.

Differentialalgebraic equations with higher index give rise to essentially ill-posed problems. Therefore, their numerical approximation requires special care. In the present paper, we state the notion of ill-posedness for linear differentialalgebraic equations more precisely. Based on this property, we construct a regularization procedure using a least-squares collocation approach by discretizing the pre-image space. Numerical experiments show that the resulting method has excellent convergence properties and is not much more computationally expensive than standard collocation methods used in the numerical solution of ordinary differential equations or index-1 differentialalgebraic equations. Convergence is shown for a limited class of linear higher-index differentialalgebraic equations.

A new differential quadrature method based on cubic B-spline is developed for the numerical solution of differential equations. In order to develop the new approach, the B-spline basis functions are used on the grid and midpoints of a uniform partition. Some error bounds are obtained by help of cubic spline collocation, which show that the method in its classic form is second order convergent. In order to derive higher accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. A new fourth order method is developed for the numerical solution of systems of second order ordinary differential equations. By solving some test problems, the performance of the proposed methods is examined. Also the implementation of the method for multi-dimensional time dependent partial differential equations is presented. The stability of the proposed methods is examined via matrix analysis. To demonstrate the applicability of the algorithms, we solve the 2D and 3D coupled Burgers’ equations and 2D sine-Gordon equation as test problems. Also the coefficient matrix of the methods for multi-dimensional problems is described to analyze the stability.