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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

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.

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.

Many complex systems in the real world evolve with time. These dynamic systems are often modeled by ordinary differential equations in mathematics. The inverse problem of ordinary differential equations is to convert the observed data of a physical system into a mathematical model in terms of ordinary differential equations. Then the model may be used to predict the future behavior of the physical system being modeled. Genetic programming has been taken as a solver of this inverse problem. Similar to genetic programming, gene expression programming could do the same job since it has a similar ability of establishing the model of ordinary differential systems. Nevertheless, such research is seldom studied before. This paper is one of the first attempts to apply gene expression programming for solving the inverse problem of ordinary differential equations. Based on a statistic observation of traditional gene expression programming, an improvement is made in our algorithm, that is, genetic operators should act more often on the dominant part of genes than on the recessive part. This may help maintain population diversity and also speed up the convergence of the algorithm. Experiments show that this improved algorithm performs much better than genetic programming and traditional gene expression programming in terms of running time and prediction precision.

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.

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.

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.

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).

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.

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.

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 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.

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.

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.

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.

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.

Our focus is the development and implementation of a new two-step hybrid method for the direct solution of general second order ordinary differential equation. Power series is adopted as the basis function in the development of the method and the arising differential system of equations is collocated at all grid and off-grid points. The resulting equation is interpolated at selected points. We then analyzed the resulting scheme for its basic properties. Numerical examples were taken to illustrate the efficiency of the method. The results obtained converge closely with the exact solutions.

In this paper we propose an efficient and easy-to-implement numerical method for an $α$-th order Ordinary Differential Equation (ODE) when $α∈ (0, 1)$, based on a one-point quadrature rule. The quadrature point in each sub-interval of a given partition with mesh size $h$ is chosen judiciously so that the degree of accuracy of the quadrature rule is 2 in the presence of the singular integral kernel. The resulting time-stepping method can be regarded as the counterpart for fractional ODEs of the well-known mid-point method for 1st-order ODEs. We show that the global error in a numerical solution generated by this method is of the order $\mathcal{O}(h^{2})$, independently of $α$. Numerical results are presented to demonstrate that the computed rates of convergence match the theoretical one very well and that our method is much more accurate than a well-known one-step method when $α$ is small.

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.

The paper analyzes the theoretical aspects of constructing a family of single-stage multi-step methods for solving the Cauchy problem with prehistory for ordinary differential equations. The authors consider general issues related to discretization, approximation, convergence, and stability. The problem of improving the accuracy of numerical solutions is analyzed in detail. The results presented in the paper are also applicable for the numerical solution of partial differential equations.

A general one-step three-hybrid (off-step) points block method is proposed for solving fourth-order initial value problems of ordinary differential equations directly. A power series approximate function is employed for deriving this method. The approximate function is interpolated atxn,xn+r,xn+s,xn+twhile its fourth and fifth derivatives are collocated at all pointsxi,i=0,r,s,t,1, in the interval of approximation. Several fourth-order initial value problems of ordinary differential equations are then solved to compare the performance of the proposed method with the derived methods. The analysis of the method reveals that the method is consistent and zero stable concluding that the method is also convergent. The numerical results demonstrate the superiority of the new method over the existing ones in terms of error.

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.

A direct time integration method is presented for the solution of systems of first order ordinary differential equations, which represent semi-discrete diffusion equations. The proposed method is based on the principle of the analog equation, which converts the N coupled equations into a set of N single term uncoupled first order ordinary differential equations under fictitious sources. The solution is obtained from the integral representation of the solution of the substitute single term equations. The stability and convergence of the numerical scheme is proved. The method is simple to implement. It is self-starting, unconditionally stable, accurate, while it does not exhibit numerical damping. The stability does not demand symmetrical and positive definite coefficient matrices. This is an important advantage, since the scheme can solve semi-discrete diffusion equations resulting from methods that do not produce symmetrical matrices, e.g. the boundary element method. The method applies also to equations with variable coefficients as well as to nonlinear ones. It performs well when long time durations are considered and it can be used as a practical method for integration of stiff parabolic equations in cases where widely used methods may fail. Numerical examples, including linear as well as non linear systems, are treated by the proposed method and its efficiency and accuracy are demonstrated.

We introduce a multiple interval Chebyshev-Gauss-Lobatto spectral collocation method for the initial value problems of the nonlinear ordinary differential equations (ODES). This method is easy to implement and possesses the high order accuracy. In addition, it is very stable and suitable for long time calculations. We also obtain thehp-version bound on the numerical error of the multiple interval collocation method underH1-norm. Numerical experiments confirm the theoretical expectations.

An efficient condition is established ensuring that on any interval of length ω, any nontrivial solution of the equation u ′′ = p ⁢ ( t ) ⁢ u ${u^{\prime\prime}=p(t)u}$ has at most one zero. Based on this result, the unique solvability of a periodic boundary value problem is studied.

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.

By the use of geometric methods for linearizing systems of second-order cubically semi-linear ordinary differential equations and the conditional linearizability of third-order quintically semi-linear ordinary differential equations, we extend to the fourth-order by differentiating the third-order conditionally linearizable equation. This yields criteria for conditional linearizability of a class of fourth-order semi-linear ordinary differential equations, which have not been discussed in the literature previously.

A new numerical method for solving fourth order ordinary differential equations directly is proposed in this paper. Interpolation and collocation were employed in developing this method using seven steps. The use of the approximated power series as an interpolation equation was adopted in deriving the method. The basic properties of the new method such as zero-stability, consistency, convergence and order are established. The numerical results indicate that the new method gives better accuracy than the existing methods when it is applied to fourth ordinary differential equations.

Departing from a numerical method designed to solve ordinary differential equations, in this manuscript we extend such approach to solve problems involving fuzzy partial differential equations. The method proposed in this work is a non-recursive technique that combines differential quadrature rules and a Picard-like scheme in order to obtain general solutions of systems of partial differential equations derived from a general fuzzy partial differential model. The property of stability and a bound for the Hausdorff distance are established under suitable conditions. Several numerical examples are provided in order to show the effectiveness of the proposed technique.