Multistep Collocation Methods Research Articles

A multistep collocation method for approximate solution of Volterra integro-differential equations of the third kind

A multistep collocation method for approximate solution of Volterra integro-differential equations of the third kind

High-order convergence of multistep collocation methods for nonstandard Volterra integral equations

Some physical and biological phenomena for the types of population growth can be modelled by nonstandard Volterra integral equations (NVIEs). In this paper, we consider multistep collocation methods to solve nonstandard Volterra integral equation and try to obtain the optimal convergence order of the numerical method. Using the fixed number of previous time steps and Lagrange basis functions, we increase the order of convergence in comparing the classical one-step collocation method. Finally, Some numerical examples are solved by multistep collocation method to illustrate the theoretical results.

Computational solution of advanced stochastic time-delay differential equations using hybrid block extended Adams Moulton methods for customer’s satisfaction when using electronic payment systems in Nigerian banks

This study examined the implementation of Hybrid Block Extended Adams Moulton Methods (HBEAMM) for the computational solution of Advanced Stochastic Time-Delay Differential Equations (ASTDDEs) through the use of various electronic payment systems such as ATM, POS and internet banking offered by the Nigerian banks. This work attempts to proffer solution to customer challenges in using the ATM of most banks in Nigeria. The challenges are considered in this work as capable of resulting to advanced time-delay and volatility noise. It is therefore modeled as advanced stochastic time-delay differential equation (ASTDDE) and solved using Hybrid Extended Block Adams Moulton Methods (HEBAMMs). The discrete schemes of the proposed method (HBEAMM) were obtained by continuous formulations of multistep collocation method by matrix inversion approach. The necessary and sufficient conditions for convergence and stability of the method were analyzed and proved satisfactory. The discrete schemes obtained were applied in solving some advanced stochastic time-delay differential equations and the results obtained revealed the degree of customers’ satisfaction in the use of e-payment systems. The accuracy of the method is compared with other existing methods and presented graphically to prove its superiority.

A multistep collocation method for solving advection equations with delay

AbstractAdvection equations with delay are appeared in the modeling of the dynamics of structured cell populations. In this article, we construct an efficient two‐dimensional multistep collocation method for the numerical solution of a class of advection equations with delay. Equations with aftereffect and equations with both aftereffect and retardation of a state variable are considered. Computability of the algorithm and convergence properties of the proposed numerical method are analyzed for solutions in appropriate Sobolev spaces, and it is shown that the proposed scheme enjoys the spectral accuracy. Numerical examples are given and comparison with other existing methods in the literature is made to demonstrate the efficiency, superiority and high accuracy of the presented method.

Multistep collocation methods for integral-algebraic equations with non-vanishing delays

In this article, we study the piecewise multistep collocation method for a class of functional integral equations with non-vanishing delays. Based on the notions of the tractability index and the ν-smoothing property of a Volterra integral operator, our numerical analysis and optimal convergence properties are investigated. Here, multistep collocation method which depends on the numerical solution in a fixed number of previous time steps is described by the constructive technique and dividing the definition domain into several subintervals according to the primary discontinuous points associated with the delay function. Numerical experiments confirm the theoretical expectations.

Implementation of Two-step Hybrid Block Adams Moulton Solution Methods for First Order Delay Differential Equations

In this paper, Hybrid Block Adams Moulton Methods for step number k = 2 merged with two and three off-grid points were obtained and implemented in solving first order delay differential equations without the use of interpolation condition in evaluating the delay expression. The discrete schemes of these off-grid hybrid block methods were assessed through the continuous development of the linear multistep collocation method using a matrix conversion formula. The results obtained after the implementation of the proposed method in for numerical experiment of some first-order DDEs, the BHAMM2 schemes performed better and faster in satisfying the axioms for convergence and region of absolute stability than the BHAMM3 schemes at fixed step size z when examined with other existing methods.

Stability analysis of the high-order multistep collocation method for the functional integral equations with constant delays

The results on the stability of recurrences play an important role in the theory of dynamical systems and computer science in connection to the notions of shadowing and controlled chaos. In this paper, stability properties of high-order multistep collocation method for functional integral equations of Volterra integral equations with constant delays type with respect to significant test equations are investigated.

On the numerical solutions for nonlinear Volterra-Fredholm integral equations

In this note, we study a class of multistep collocation methods for the numerical integration of nonlinear Volterra-Fredholm Integral Equations (V-FIEs). The derived method is characterized by a lower triangular or diagonal coefficient matrix of the nonlinear system for the computation of the stages which, as it is known, can beexploited to get an efficient implementation. Convergence analysis and linear stability estimates are investigated. Finally numerical experiments are given, which confirm our theoretical results.

Multistep collocation method for Fredholm integral equations of the second kind

In this paper a Multistep Collocation (MC) method for the second kind Fredholm integral equations (FIEs) is proposed and analyzed. The multistep collocation method is applied to FIE with smooth kernels under uniform mesh and weakly singular kernels |s−t|−α(0<α<1) using a graded mesh then the same convergence rate as collocation method but with a lower degree of freedom is obtained. Moreover, in order to avoid the round-off errors caused by graded mesh, a Hybrid Multistep Collocation (HMC) method by combining multistep collocation and hybrid collocation method is proposed. The HMC method converges faster with lower degrees of freedom and more efficiently captures the weakly singular properties by nonpolynomial interpolation at the first subinterval. The L∞-norm convergence results are analysed and proved. Numerical examples are presented to demonstrate the efficiency of the proposed methods.

Solving the third-kind Volterra integral equation via the boundary value technique: Lagrange polynomial versus fractional interpolation

The solution to the third-kind Volterra integral equation (VIE3) usually has unbounded derivatives near the original point t=0, which brings difficulties to numerical computation. In this paper, we analyze two kinds of modified multistep collocation methods for VIE3: collocation boundary value method with the fractional interpolation (FCBVM) and that with Lagrange interpolation (CBVMG). The former is developed based on the non-polynomial interpolation which is particularly feasible for approximating functions in the form of tη with the real number η≥0. The latter is devised by using classical polynomial interpolation. The application of the boundary value technique enables both approaches to efficiently solve long-time integration problems. Moreover, we investigate the convergence properties of these two kinds of algorithms by Grönwall’s inequality.

Numerical treatment of first order delay differential equations using extended block backward differentiation formulae

In this research, we developed and implemented extended backward differentiation methods (formulae) in block forms for step numbers k = 2, 3 and 4 to evaluate numerical solutions for certain first-order differential equations of delay type, generally referred to as delay differential equations (DDEs), without the use of interpolation methods for estimating the delay term. The matrix inversion approach was applied to formulate the continuous composition of these block methods through linear multistep collocation method. The discrete schemes were established through the continuous composition for each step number, which evaluated the error constants, order, consistency, convergent and area of absolute equilibrium of these discrete schemes. The study of the absolute error results revealed that, as opposed to the exact solutions, the lower step number implemented with super futures points work better than the higher step numbers implemented with super future points.

Analysis of Generalized Multistep Collocation Solutions for Oscillatory Volterra Integral Equations

In this work, we introduce a class of generalized multistep collocation methods for solving oscillatory Volterra integral equations, and study two kinds of convergence analysis. The error estimate with respect to the stepsize is given based on the interpolation remainder, and the nonclassical convergence analysis with respect to oscillation is developed by investigating the asymptotic property of highly oscillatory integrals. Besides, the linear stability is analyzed with the help of generalized Schur polynomials. Several numerical tests are given to show that the numerical results coincide with our theoretical estimates.

On the convergence of multistep collocation methods for integral-algebraic equations of index 1

The multistep collocation method is introduced to solve integral-algebraic equations of index 1. The existence and uniqueness of the multistep collocation solution are proved. The convergence of the perturbed multistep collocation method is also investigated, which extends and includes the analysis of the multistep collocation method without perturbed terms. Some numerical experiments are given to illustrate the theoretical results.

Numerical analysis of a high order method for nonlinear delay integral equations

In this paper, we introduce a new class of Multistep Collocation Method (MCM) for solving the non-linear Volterra Functional Integral Equations (VFIEs) including two types of linear and non-linear lag functions. We divide the definition domain into several subintervals according to the primary discontinuous points associated with the delay function. In each subinterval, where the solution is smooth enough, we can apply MCM to approximate the solution. Convergence analysis of the method for both types is also analyzed and illustrative five examples are provided to confirm the applicability of this method.

A class of fifth order block hybrid Adams Moultonrsquo;s method for solving stiff initial value problems

This paper is concern with the derivation of the continuous formulations of hybrid Adams-Moulton Method for step number and incorporating three, two and one off-grid points respectively by multistep collocation method using matrix inversion technique. The discrete schemes used in block form were derived from their respective continuous formulations. The convergence analysis were carried and found that the methods are convergent. The region of absolute stability of the methods were plotted in a block form and they are all - stable. Some linear stiff problems were solved using the schemes. It was observed that the schemes for step number incorporating one off-grid point perform better than the schemes for step number incorporating two and three off-grid points respectively.

A numerical method for solving Volterra integral equations of the third kind by multistep collocation method

In this paper a multistep collocation method for solving Volterra integral equations of the third kind is explained and analyzed. The structure of the method, its solvability and convergence analysis are investigated. Moreover to show the applicability of the presented method and to confirm our theoretical results some numerical examples are given.

Multistep Collocation Methods for Volterra Integral Equations with Weakly Singular Kernels

Multistep Collocation Methods for Volterra Integral Equations with Weakly Singular Kernels

Fractional retarded differential equations and their numerical solution via a multistep collocation method

In this paper, we consider the nonlinear fractional retarded differential equations (FRDE). We extend the results of the existence and uniqueness of the solution, the propagation of derivative discontinuities and the dependence of the solution on the parameters of the equation. Next, we develop an efficient multistep collocation method for solving this type of equations. The proposed scheme is especially suited for FRDEs with piecewise smooth solutions, due to its essential feature of local approximations on subintervals. The stability of the scheme is accessed, and the convergence analysis is studied for functions in appropriate Sobolev spaces. Numerical results confirm the spectral accuracy and the stability of the proposed method for large domain calculations.

A collocation method for differential equations with one‐point Taylor initial conditions

This paper presents a new multistep collocation method for nth order differential equations. The interval of interest is first divided into N subintervals. By determining interval conditions in Taylor interpolation in each subinterval, Taylor polynomials are calculated with different step lengths. Then the obtained solutions in each subinterval are used as initial conditions in the next step. Optimal error is assessed using Peano lemma, which shows that the errors decay by decreasing the step length. In particular, for fixed step length h, the error is of O(m−m), where m is the number of collocation points in each subinterval. Meanwhile, for fixed m, the error is of O(hm). Numerical examples demonstrate the validity and high accuracy of the proposed method even for stiff problems.

Collocation Methods for Volterra Integral and Integro-Differential Equations: A Review

We present a collection of recent results on the numerical approximation of Volterra integral equations and integro-differential equations by means of collocation type methods, which are able to provide better balances between accuracy and stability demanding. We consider both exact and discretized one-step and multistep collocation methods, and illustrate main convergence results, making some comparisons in terms of accuracy and efficiency. Some numerical experiments complete the paper.