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The Smooth Composite Chebyshev Finite Difference method is generalized for higher order initial and boundary value problems. Round-off and truncation error analyses and convergence analysis of the method are also extended to higher order. The proposed method is applied to obtain the highly precise numerical solutions of boundary or initial value problems of the Bratu and higher order Lane Emden types. To visualize the competency of the presented method, the obtained results are compared with nine different methods, namely, Bezier curve method, Adomian decomposition method, Operational matrix collocation method, Direct collocation method, Haar Wavelet Collocation, Bernstein Collocation Method, Improved decomposition method, Quartic B-Spline method and New Cubic B-spline method. The comparisons show that the presented method is highly accurate than the other numerical methods and also gets rid of the singularity of the given problems. [ABSTRACT FROM AUTHOR]

Highlights • Applications of the proposed BSLM in nonlinear differential equations are discussed. • The possibility to apply the method in the KP equations is suggested. Abstract In this study, we develop a numerical method for solving the coupled viscous Burgers' equations based on a backward semi-Lagrangian method. The main difficulty associated with the backward semi-Lagrangian method for this problem is treating the nonlinearity in the diffusion-reaction equation, whose reaction coefficients are given in terms of coupled partial derivatives. To handle this difficulty, we use an extrapolation technique which splits the nonlinearity into two linear diffusion-reaction boundary value problems. In the proposed backward semi-Lagrangian method, we use fourth-order finite differences to discretize the diffusion-reaction boundary value problems and employ the so-called error correction method to solve the highly nonlinear initial value problems. Our overall algorithm is completely iteration-free and computationally efficient. We demonstrate the numerical accuracy and efficiency of the present method by comparing our numerical results with analytical solutions and other numerical solutions based on alternative existing methods. [ABSTRACT FROM AUTHOR]

A procedure is given for continuously computing and monitoring the step size to be used by a self-starting, pth-order numerical integration method to solve an initial value problem. The procedure uses an estimate of the truncation error to calculate the step size. [ABSTRACT FROM AUTHOR]

A well-known drawback of algorithms based on Taylor series formulae is that the explicit calculation of higher order derivatives formally is an over-elaborate task. To avoid the analytical computation of the successive derivatives, numeric and automatic differentiation are usually used. A recent alternative to these techniques is based on the calculation of higher derivatives by using the Infinity Computer—a new computational device allowing one to work numerically with infinities and infinitesimals. Two variants of a one-step multi-point method closely related to the classical Taylor formula of order three are considered. It is shown that the new formula is order three accurate, though requiring only the first two derivatives of y ( t ) (rather than three if compared with the corresponding Taylor formula of order three). To get numerical evidence of the theoretical results, a few test problems are solved by means of the new methods and the obtained results are compared with the performance of Taylor methods of order up to four. [ABSTRACT FROM AUTHOR]

Based on the truncated Painlevé method or the Möbious (conformal) invariant form, the nonlocal symmetry for the (1+1)–dimensional Gardner equation is derived. The nonlocal symmetry can be localized to the Lie point symmetry by introducing one new dependent variable. Thanks to the localization procedure, the finite symmetry transformations are obtained by solving the initial value problem of the prolonged systems. Furthermore, by using the symmetry reduction method to the enlarged systems, many explicit interaction solutions among different types of solutions such as solitary waves, rational solutions, Painlevé II solutions are given. Especially, some special concrete soliton-cnoidal interaction solutions are analyzed both in analytical and graphical ways. [ABSTRACT FROM AUTHOR]

We consider traveling wave solutions to the Casimir equation for the Ito system (a two-field extension of the KdV equation). These traveling waves are governed by a nonlinear initial value problem with an interesting nonlinearity (which actually amplifies in magnitude as the size of the solution becomes small). The nonlinear problem is parameterized by two initial constant values, and we demonstrate that the existence of solutions is strongly tied to these parameter values. For our interests, we are concerned with positive, bounded, periodic wave solutions. We are able to classify parameter regimes which admit such solutions in full generality, thereby obtaining a nice existence result. Using the existence result, we are then able to numerically simulate the positive, bounded, periodic solutions. We elect to employ a group preserving scheme in order to numerically study these solutions, and an outline of this approach is provided. The numerical simulations serve to illustrate the properties of these solutions predicted analytically through the existence result. Physically, these results demonstrate the existence of a type of space-periodic structure in the Casimir equation for the Ito model, which propagates as a traveling wave. [ABSTRACT FROM AUTHOR]

This paper is devoted to introduce a new concept of upper and lower solutions for fuzzy integro-differential equations, that is defined on integral equations associated with the main problem. This plays an essential role in proof of existence and uniqueness results under weak conditions. By the method of upper and lower solutions, existence and uniqueness of global solutions of an initial value problem for first-order nonlinear fuzzy integro-differential equation under generalized differentiability are given. Our technique is illustrated by some examples. [ABSTRACT FROM AUTHOR]

In this work we derive new alternatives for efficient computation of the matrix cosine which is useful when solving second order Initial Value Problems such as free vibration. We focus especially on the two classes of normal and nonnegative matrices and we present intervals of applications for rational L ∞ approximations of various degrees for these types of matrices in the lines of Hargreaves and Higham (2005). Our method relies on Remez algorithm for rational approximation while the innovation here is the choice of the starting set of non-symmetrical Chebyshev points. Only one Remez iteration is then usually enough to quickly approach the actual L ∞ approximant. [ABSTRACT FROM AUTHOR]

Highlights: [•] Methods based on series approximations are useful for differential equations. [•] Projections in a Hilbert space basis, combined with optimization algorithms, provide precise solutions. [•] Dirac’s formalism allows extensions for complex functions. [•] Fourier decomposition provides a simple and precise algorithm to solve differential equations. [•] Precision is considerably improved by using this kind of method. [ABSTRACT FROM AUTHOR]

Abstract: In this article, we establish existence results of at least one solution for two classes of initial value problems of nonlinear impulsive fractional differential models on half lines involving Caputo fractional derivatives. Our analysis rely on the well known fixed point theorem of Schauder and nonlinear alternative of Leray–Schauder type. [Copyright &y& Elsevier]

Abstract: In this paper, we review the so-called Töpfer algorithm that allows us to find a non-iterative numerical solution of the Blasius problem, by solving a related initial value problem and applying a scaling transformation. Moreover, we remark that the applicability of this algorithm can be extended to any given problem, provided that the governing equation and the initial conditions are invariant under a scaling group of point transformations and that the asymptotic boundary condition is non-homogeneous. Then, we describe an iterative extension of Töpfer’s algorithm that can be applied to a general class of problems. Finally, we solve the Falkner–Skan model, for values of the parameter where multiple solutions are admitted, and report original numerical results, in particular data related to the famous reverse flow solutions by Stewartson. The numerical data obtained by the extended algorithm are in good agreement with those obtained in previous studies. [Copyright &y& Elsevier]

Abstract: In this paper, the initial value problems of fuzzy differential systems are investigated by using a kind of new methods. At first, the concept of relatively open sets is introduced. Then, some basic topological properties of solutions for the initial value problems of fuzzy differential systems are obtained. [Copyright &y& Elsevier]

Abstract: The applicability of the Parareal parallel-in-time integration scheme for the solution of a linear, two-dimensional hyperbolic acoustic-advection system, which is often used as a test case for integration schemes for numerical weather prediction (NWP), is addressed. Parallel-in-time schemes are a possible way to increase, on the algorithmic level, the amount of parallelism, a requirement arising from the rapidly growing number of CPUs in high performance computer systems. A recently introduced modification of the “parallel implicit time-integration algorithm” could successfully solve hyperbolic problems arising in structural dynamics. It has later been cast into the framework of Parareal. The present paper adapts this modified Parareal and employs it for the solution of a hyperbolic flow problem, where the initial value problem solved in parallel arises from the spatial discretization of a partial differential equation by a finite difference method. It is demonstrated that the modified Parareal is stable and can produce reasonably accurate solutions while allowing for a noticeable reduction of the time-to-solution. The implementation relies on integration schemes already widely used in NWP (RK-3, partially split forward Euler, forward–backward). It is demonstrated that using an explicit partially split scheme for the coarse integrator allows to avoid the use of an implicit scheme while still achieving speedup. [Copyright &y& Elsevier]

Abstract: In this work a procedure for the construction of an explicit optimized Runge–Kutta–Nyström method with four stages and fifth algebraic order is provided. The variable coefficients of the preserved method result after nullifying the phase-lag, the dissipative error and the first derivative of the phase-lag. We can see the efficiency of the new method through its local truncation error. Furthermore, we compare the new methodʼs efficiency to other numerical methods. This is shown through the integration of the two-body problem with various eccentricities and of four other initial value problems. [Copyright &y& Elsevier]

Abstract: We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves. We study the dynamics of weakly nonlinear waves, and using asymptotic multiple-scale expansions and averaging with respect to the fast time, we obtain a hierarchy of asymptotically exact coupled and uncoupled Ostrovsky equations for unidirectional waves. We then construct a weakly nonlinear solution of the initial-value problem in terms of solutions of the derived Ostrovsky equations within the accuracy of the governing equations, and show that there are no secular terms. When coupling parameters are equal to zero, our results yield a weakly nonlinear solution of the initial-value problem for the Boussinesq equation in terms of solutions of the initial-value problems for two Korteweg–de Vries equations, integrable by the Inverse Scattering Transform. We also perform relevant numerical simulations of the original unapproximated system of Boussinesq equations to illustrate the difference in the behaviour of its solutions for different asymptotic regimes. [Copyright &y& Elsevier]

Abstract: Caputo q-fractional derivatives are introduced and studied. A Caputo -type q-fractional initial value problem is solved and its solution is expressed by means of a new introduced q-Mittag–Leffler function. Some open problems about q-fractional integrals are proposed as well. [Copyright &y& Elsevier]

Abstract: A stochastic predator–prey system with Holling II functional response is proposed and investigated. We show that there is a unique positive solution to the model for any positive initial value. And we show that the positive solution to the stochastic system is stochastically bounded. Moreover, under some conditions, we conclude that the stochastic model is stochastically permanent and persistent in mean. [Copyright &y& Elsevier]

Abstract: The present paper deals with an optimal boundary control problem in which the process of systems under consideration is governed by a linear parabolic partial differential equation over an infinite time interval. The objective of the paper is to determine the optimal boundary control that minimize a given energy-based performance measure. The performance measure is specified as a quadratic functional of displacement and a suitable penalty term involving the boundary controls. In order to determine the optimal boundary controls, the problem with boundary controls are converted into a problem with distributed controls. The modal space technique is then used to reduce the system into the optimal control of time invariant lumped parameter system. The associated system of uncoupled first order initial value problems is solved in terms of controllers. Next step deals with the computation of the control and trajectory of the linear time-invariant lumped parameter. For this we approximate the controllers by a finite number of orthogonal exponential zero-interpolants over the interval [0,∞). The resultant performance index after using the optimality condition leads to a system of linear algebraic equations. The suggested technique is easy to implement on digital computer. We provide a numerical example to demonstrate the applicability and efficiency of the proposed approach. [Copyright &y& Elsevier]

Abstract: The Adomian decomposition method (ADM) cannot deal generally with the non-homogenous differential equations. The method still needs improvements. This paper introduces a qualitative improvement in the method. The improved method is called the improved Adomian decomposition method (IADM). A comparison between ADM and IADM shows that the ADM gives results that is not accepted in some examples and cannot solve the other examples while IADM solves the problems effectively and efficiently. [Copyright &y& Elsevier]

Abstract: The unusual implementation of explicit Falkner methods for solving special second-order initial-value problems increases the order of the method as will be shown by means of numerical examples. In this paper we made an analysis of the propagation of the truncation errors in both the usual and the unusual implementations, thus justifying the numerical results obtained. By that, some error bounds for the global truncation errors on the solution and on the derivative are provided. Some numerical examples confirm that the bounds are realistic. Stability analysis is also addressed, and intervals of stability are presented. [Copyright &y& Elsevier]

Abstract: In this paper, the variational iteration method (VIM) is applied to solve singular perturbation initial value problems (SPIVPs). The obtained sequence of iterates is based on the use of Lagrange multipliers. Some convergence results of VIM for solving SPIVPs are given. Moreover, the illustrative examples show the efficiency of the method. [Copyright &y& Elsevier]

Abstract: In this work, we propose a novel computational algorithm for solving linear and nonlinear initial value problems by using the modified version of differential transform method (DTM), which is called the projected differential transform method (PDTM). The PDTM can be easily applied to the initial value problems with less computational work. For the several illustrative examples, the computational results are compared with those obtained by many other methods; the Adomian decomposition, the variational iteration and the spline method. For all examples, the PDTM provides exact solutions. It has been shown that the PDTM is a reliable algorithm in obtaining analytic as well as approximate solution for the initial value problems. [Copyright &y& Elsevier]

Abstract: When we construct continuous and/or discrete mathematical models in order to describe a real-life problem, these models should have various qualitative properties, which typically arise from some basic principles of the modelled phenomena. In this paper we investigate this question for the numerical solution of initial-boundary problems for the parabolic problems in higher dimensions, with the first boundary condition, using the linear finite elements. We give the conditions for the geometry of the mesh and for the choice of the discretization parameters, i.e., for the step sizes under which the discrete qualitative properties hold. For the special regular uniform simplicial mesh we define the conditions for the discretization step-sizes. [Copyright &y& Elsevier]

Abstract: For inconsistent initial value problem of singular linear system, passing from classical solution to distributional solution is formulated as a process of embedding the system with initial condition into distribution space. Three technical notions, function space on with given initial value, consistent extension of the pair of the function space and input, and order exchange of the two operations of embedding and differentiating, result in a rigorous definition of distributional solution. [Copyright &y& Elsevier]

Abstract: The particular approximate solution of the initial boundary valued problem to the Cahn–Hilliard equation is provided. The Fourier Method is combined with the Adomian’s decomposition method in order to provide an approximate solution that satisfy the initial and the boundary conditions. The approximate solution also satisfies the mass conservation principle. [Copyright &y& Elsevier]

Abstract: Two important models for rigid-plastic flows of granular materials are the non-associated plastic potential flow rule and the double shearing model. Both models generalise the equations governing metal plasticity and both, unlike the well-posed metal plasticity equations, are ill-posed in the sense that solutions to initial value problems do not depend continuously on the initial data. For the non-associated plastic potential model ill-posedness is a consequence of the non-coincidence of the velocity and stress characteristics, whereas for the double shearing model the presence of the material time rate of change of the inclination of the major principal axis of stress gives rise to a spatially elliptic system. A recently proposed model represents a generalisation of both the plastic potential and double shearing model and which incorporates a non-coaxial term in which the stress rotation rate of the double shearing model is replaced by a Cosserat intrinsic spin. The new model preserves the characteristic structure of metal plasticity, i.e. the characteristics exist and the non-coaxial term can be chosen so as to give rise to coincident stress and velocity characteristics. [Copyright &y& Elsevier]

Abstract: The effectiveness of a semi-implicit (SI) temporal scheme is discussed in the context of the dispersive magnetohydrodynamics where, due to the whistler modes, stability of explicit algorithms requires a time step decreasing quadratically as the resolution is linearly increased. After analyzing the effects of this scheme on the Alfvén-wave dispersion relation, spectral simulations of nonlinear initial value problems where small-scale dispersion has a main effect on the global dynamics are presented. Permitting a moderate, albeit significant, increase of the time step for a minor additional cost relatively to explicit schemes, the SI algorithm provides an efficient tool in situations, such as turbulent regimes, where the time steps making fully implicit schemes efficient are too large to ensure a satisfactory accuracy. [Copyright &y& Elsevier]

Abstract: A new Runge–Kutta–Nyström method, with phase-lag of order infinity, for the integration of second-order periodic initial-value problems is developed in this paper. The new method is based on the Dormand, El-Mikkawy and Prince Runge–Kutta–Nyström method of algebraic order four with four (three effective) stages. Numerical illustrations indicate that the new method is much more efficient than other methods derived, based on the idea of minimal phase lag or of phase lag of order infinity. [Copyright &y& Elsevier]

Abstract: In this paper, the class of linear generalized neutral differential delay systems with time-invariant coefficients is studied. These kinds of systems are inherent in many physical and engineering phenomena. Using the matrix pencil theory, we decompose it into five subsystems, whose solutions are obtained. Moreover, the form of the initial function is given, so the corresponding initial value problem is uniquely solvable. [Copyright &y& Elsevier]

Abstract: In this paper, some numerical procedures for solving fuzzy first-order initial value problem have been investigated. Sufficiently conditions for stability and convergence of the proposed algorithms are given and their applicability is illustrated with examples. [Copyright &y& Elsevier]

Abstract: In this paper, an improved predictor–corrector (IPC) method is proposed to solve the “fuzzy initial value problem”. The IPC method is generated by combining an explicit three-step method and an implicit two-step method. The methods are compared with the methods discussed in [T. Allahviranloo, N. Ahmady, E. Ahmady, Numerical solution of fuzzy differential equations by predictor–corrector method, Information Sciences 177(7) (2007) 1633–1647], and are shown to be more accurate. The convergence and stability of the proposed methods are also presented in detail. These methods are illustrated by solving some examples. [Copyright &y& Elsevier]

Abstract: Finite precision computations may affect the stability of algorithms and the accuracy of computed solutions. In this paper we first obtain a relation for computing the number of common significant digits between the exact solution and a computed solution of a one-dimensional initial-value problem obtained by using a single-step or multi-step method. In fact, by using the approximate solutions obtained with stepsizes h and h /2, the number of common significant digits between approximate solution with stepsize h and exact solution is estimated. Then by using the stochastic arithmetic, the CESTAC method, and the CADNA library we propose an algorithm to control the round-off error effect on the computed solution. This method can easily apply to a system of n one-dimensional initial-value problems. Finally some numerical examples are given to show the efficiency of the method. [Copyright &y& Elsevier]

Abstract: In a recent paper, Shokri et al. (2011) [1] introduce a new class of hybrid Obrechkoff methods for the numerical solution of second-order initial value problems. In this work, we will derive -stable hybrid Obrechkoff methods for the numerical integration of stiff first-order initial value problems. The numerical results obtained by the new method for some problems show its superiority in efficiency, accuracy and stability. [Copyright &y& Elsevier]