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In this paper, we investigate the numerical approximation of stochastic convection-reaction-diffusion equations using two explicit exponential integrators. The stochastic partial differential equation (SPDE) is driven by additive Wiener process. The approximation in space is done via a combination of the standard finite element method and the Galerkin projection method. Using the linear functional of the noise, we construct two accelerated numerical methods, which achieve higher convergence orders. In particular, we achieve convergence rates approximately $1$ for trace class noise and $\frac{1}{2}$ for space-time white noise. These convergences orders are obtained under less regularities assumptions on the nonlinear drift function than those used in the literature for stochastic reaction-diffusion equations. Numerical experiments to illustrate our theoretical results are provided

In this paper, we study the numerical approximation of a general second order semilinear stochastic partial differential equation (SPDE) driven by a additive fractional Brownian motion (fBm) with Hurst parameter $H>\frac {1}{2}$ H > 1 2 and Poisson random measure. Such equations are more realistic in modelling real world phenomena. To the best of our knowledge, numerical schemes for such SPDE have been lacked in the scientific literature. The approximation is done with the standard finite element method in space and three Euler-type timestepping methods in time. More precisely the well-known linear implicit method, an exponential integrator and the exponential Rosenbrock scheme are used for time discretization. In contract to the current literature in the field, our linear operator is not necessary self-adjoint and we have achieved optimal strong convergence rates for SPDE driven by fBm and Poisson measure. The results examine how the convergence orders depend on the regularity of the noise and the initial data and reveal that the full discretization attains the optimal convergence rates of order $\mathcal {O}(h^{2}+\varDelta t)$ O ( h 2 + Δ t ) for the exponential integrator and implicit schemes. Numerical experiments are provided to illustrate our theoretical results for the case of SPDE driven by the fBm noise.

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

We consider the numerical approximation of the general second order semilinear parabolic stochastic partial differential equations (SPDEs) driven by additive space–time noise. Our goal is to build two numerical algorithms with strong convergence rates higher than that of the standard semi-implicit scheme. In contrast to the standard time stepping methods which use basic increments of the noise, we introduce two schemes based on the exponential integrators, designed for finite element, finite volume or finite difference space discretisations. We prove the convergence in the root mean square L 2 norm for a general advection diffusion reaction equation and a family of new Lipschitz nonlinearities. We observe from both the analysis and numerics that the proposed schemes have better convergence properties than the current standard semi-implicit scheme.

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

High-order uncertain differential equations are used to model differentiable uncertain systems with high-order differentials, and how to solve the high-order uncertain differential equations is a core issue in practice. This paper aims at proposing a numerical method to solve the high-order uncertain differential equations based on the Runge–Kutta recursion formula, which is of high precision degree. A procedure is designed and some numerical experiments are performed to illustrate the effectiveness and efficiency of the Runge–Kutta method. In addition, this paper also presents a numerical method to calculate the expected value of a function of the solution.

We introduce the concept of numerical Gaussian processes, which we define as Gaussian processes with covariance functions resulting from temporal discretization of time-dependent partial differential equations. Numerical Gaussian processes, by construction, are designed to deal with cases where (a) all we observe are noisy data on black-box initial conditions, and (b) we are interested in quantifying the uncertainty associated with such noisy data in our solutions to time-dependent partial differential equations. Our method circumvents the need for spatial discretization of the differential operators by proper placement of Gaussian process priors. This is an attempt to construct structured and data-efficient learning machines, which are explicitly informed by the underlying physics that possibly generated the observed data. The effectiveness of the proposed approach is demonstrated through several benchmark problems involving linear and nonlinear time-dependent operators. In all examples, we are able to r...

In this work, we take as our base scalar second-order ordinary differential equations (ODEs) which have seven equivalence classes with each class possessing three Lie point symmetries. We show how one can calculate the conditional symmetries of third-order non-linear ODEs subject to root second-order nonlinear ODEs which admit three point symmetries. Moreover, we show when scalar second-order ODEs taken as first integrals or conditional first integrals are inherited as Lie point symmetries and as conditional symmetries of the derived third-order ODE. Furthermore, the derived scalar nonlinear third-order ODEs without substitution are considered for their conditional symmetries subject to root second-order ODEs having three symmetries.

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.

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.

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.

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.

A novel mesh-free heuristic method for solving differential equations is proposed. The new approach can cope with linear, nonlinear, and partial differential equations (DE), and systems of DEs. Candidate solutions are expressed using a linear combination of kernel functions. Thus, the original problem is transformed into an optimization problem that consists in finding the parameters that define each kernel. The new optimization problem is solved applying a Covariance Matrix Adaptation Evolution Strategy. To increase the accuracy of the results, a Downhill Simplex local search is applied to the best solution found by the mentioned evolutionary algorithm. Our method is applied to 32 differential equations extracted from the literature. All problems are successfully solved, achieving competitive accuracy levels when compared to other heuristic methods. A simple comparison with numerical methods is performed using two partial differential equations to show the pros and cons of the proposed algorithm. To verify the potential of this approach with a more practical problem, an electric circuit is analyzed in depth. The method can obtain the dynamic behavior of the circuit in a parametric way, taking into account different component values.

A numerical time-stepping algorithm for differential or partial differential equations is proposed that adaptively modifies the dimensionality of the underlying modal basis expansion. Specifically, the method takes advantage of any underlying low-dimensional manifolds or subspaces in the system by using dimensionality-reduction techniques, such as the proper orthogonal decomposition, in order to adaptively represent the solution in the optimal basis modes. The method can provide significant computational savings for systems where low-dimensional manifolds are present since the reduction can lower the dimensionality of the underlying high-dimensional system by orders of magnitude. A comparison of the computational efficiency and error for this method are given showing the algorithm to be potentially of great value for high-dimensional dynamical systems simulations, especially where slow-manifold dynamics are known to arise. The method is envisioned to automatically take advantage of any potential computational saving associated with dimensionality-reduction, much as adaptive time-steppers automatically take advantage of large step sizes whenever possible.

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 the present paper we provide some existence results for the Darboux problem of partial fractional random differential equations with state-dependent delay by applying the measure of noncompactness and a random fixed point theorem with stochastic domain.

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.

We consider nonclassical boundary value problems with discontinuous conditions for some systems of partial differential equations of the first and second order.

This study presents numerical solutions to linear and nonlinear Partial Differential Equations (PDEs) by using the peridynamic differential operator. The solution process involves neither a derivative reduction process nor a special treatment to remove a jump discontinuity or a singularity. The peridynamic discretization can be both in time and space. The accuracy and robustness of this differential operator is demonstrated by considering challenging linear, nonlinear, and coupled PDEs subjected to Dirichlet and Neumann-type boundary conditions. Their numerical solutions are achieved using either implicit or explicit methods. © 2017 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1726–1753, 2017

This paper presents a method to solve the problems of solutions for integer differential and partial differential equations using the convergence of Adomian’s Method. In this paper, we firstly use the convergence of Adomian’s Method to derive the solutions of high order linear fractional equations, and then the numerical solutions for nonlinear fractional equations. we also get the solutions of two fractional reaction-diffusion equations. We can see the advantage of this method to deal with fractional differential equations.

Exponential additive Runge-Kutta methods for solving semi-linear equations are discussed. Related order conditions and stability properties for both explicit and implicit schemes are developed, according to the dimension of the coefficients in the linear terms. Several examples illustrate our theoretical results.

In the paper we study the solution and smoothness of the solution of one class of partial differential equations of higher order in bounded domain $G \subset \mathbb{R}^{n}$ satisfying the flexible $\lambda$-horn condition.

In this article, the fractional derivatives in the sense of modified Riemann-Liouville derivative and the Exp-function method are employed for constructing the exact solutions of nonlinear time fractional partial differential equations in mathematical physics. As a result, some new exact solutions for them are successfully established. It is indicated that the solutions obtained by the Exp-function method are reliable, straightforward and effective method for strongly nonlinear fractional partial equations with modified Riemann-Liouville derivative by Jumarie U+02BC s. This approach can also be applied to other nonlinear time and space fractional differential equations.

Consideration of partial mappings allows us to expose a noticeable part of the theory of ordinary differential equations at the axiomatic level. This gives us a possibility to cover equations with discontinuous right-hand sides and differential inclusions by standard well-known (but a little revised) theory. Homological properties of the corresponding solution sets allow us to create a new version of the shift method in the theory of boundary value problems as powerful as the application here of the Leray–Schauder theory.

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

We consider the problems arising in the construction of the solutions of singularly perturbed differential equations. Usually, the decision of such problems by standard methods encounters significant difficulties of various kinds. The use of a common neural network approach is demonstrated in three model problems for ordinary differential equations. The conducted computational experiments confirm the effectiveness of this approach.

We propose a new method called the fractional reduced differential transform method (FRDTM) to solve nonlinear fractional partial differential equations such as the space-time fractional Burgers equations and the time-fractional Cahn-Allen equation. The solutions are given in the form of series with easily computable terms. Numerical solutions are calculated for the fractional Burgers and Cahn-Allen equations to show the nature of solutions as the fractional derivative parameter is changed. The results prove that the proposed method is very effective and simple for obtaining approximate solutions of nonlinear fractional partial differential equations.

The study deals with nonlinear ordinary differential equations defined parametrically by two relations; these arise in fluid dynamics and are a special class of coupled differential–algebraic equations. We propose a few techniques for reducing such equations, first or second order, to systems of standard ordinary differential equations as well as techniques for the exact integration of these systems. Several examples show how to construct general solutions to some classes of nonlinear equations involving arbitrary functions. We specify a procedure for the numerical solution of the Cauchy problem for parametrically defined differential equations and related differential–algebraic equations. The proposed techniques are also effective for the numerical integration of problems for implicitly defined equations.

In this paper, we consider the numerical method that is proposed and analyzed in [J. Cao and C. Xu, J. Comput. Phys., 238 (2013), pp. 154–168] for the fractional ordinary differential equations. It is based on the so-called block-by-block approach, which is a common method for the integral equations. We extend the technique to solve the nonlinear system of fractional ordinary differential equations (FODEs) and present a general technique to construct high order schemes for the numerical solution of the nonlinear coupled system of fractional ordinary differential equations (FODEs). By using the present method, we are able to construct a high order schema for nonlinear system of FODEs of the orderα,α>0. The stability and convergence of the schema is rigorously established. Under the smoothness assumptionf,g∈C4[0,T], we prove that the numerical solution converges to the exact solution with order 3+αfor 0<α≤1 and order 4 forα>1. Some numerical examples are provided to confirm the theoretical claims.

The barycentric interpolation collocation method is discussed in this paper, which is not valid for singularly perturbed delay partial differential equations. A modified version is proposed to overcome this disadvantage. Two numerical examples are provided to show the effectiveness of the present method.

New simple and robust methods have been proposed for detecting poles, logarithmic poles, and mixed-type singularities in systems of ordinary differential equations. The methods produce characteristics of these singularities with a posteriori asymptotically precise error estimates. This approach is applicable to an arbitrary parametrization of integral curves, including the arc length parametrization, which is optimal for stiff and ill-conditioned problems. The method can be used to detect solution blowup for a broad class of important nonlinear partial differential equations, since they can be reduced to huge-order systems of ordinary differential equations by applying the method of lines. The method is superior in robustness and simplicity to previously known methods.

We consider the following two-dimensional system of non-linear equations: u ′ = g ⁢ ( t ) ⁢ | v | 1 α ⁢ sgn ⁡ v , v ′ = - p ⁢ ( t ) ⁢ | u | α ⁢ sgn ⁡ u , u^{\prime}=g(t)|v|^{\frac{1}{\alpha}}\operatorname{sgn}v,\quad v^{\prime}=-p(t% )|u|^{\alpha}\operatorname{sgn}u, where α > 0 {\alpha>0} , and g : [ 0 , + ∞ [ → [ 0 , + ∞ [ {g\colon{[0,+\infty[}\rightarrow{[0,+\infty[}} and p : [ 0 , + ∞ [ → ℝ {p\colon{[0,+\infty[}\rightarrow\mathbb{R}} are locally integrable functions. Moreover, we assume that the coefficient g is non-integrable on [ 0 , + ∞ ] {[0,+\infty]} . We establish new non-oscillation criteria for the considered system, which generalize known results for the corresponding linear system and for second order differential equations. In particular, the presented criteria are in compliance with the results of Hille and Nehari.

We consider the mean-square stability of the so-called improved split-step theta method for stochastic differential equations. First, we study the mean-square stability of the method for linear test equations with real parameters. When θ ≥ 3/2, the improved split-step theta methods can reproduce the mean-square stability of the linear test equations for any step sizes h > 0. Then, under a coupled condition on the drift and diffusion coefficients, we consider exponential mean-square stability of the method for nonlinear non-autonomous stochastic differential equations. Finally, the obtained results are supported by numerical experiments.

In this paper, Haar wavelet collocation method is applied to obtain the numerical solution of a particular class of delay differential equations. The method is applied to linear and nonlinear delay differential equations as well as systems involving these delay differential equations. In addition to this the method is also extended to numerical solution of delay partial differential equations with delay in time. The method is applied to several benchmark test problems. The numerical results are compared with the exact solutions and the performance of the method is demonstrated by calculating the maximum absolute errors and experimental rates of convergence using different numbers of collocation points. The numerical results show that the method is simply applicable, accurate, efficient and robust.