An effective computational approach based on new multi-step collocation method for solving integral-algebraic equations

Multistep collocation method for Fredholm integral equations of the second kind

Multistep collocation approximations to solutions of first-kind Volterra integral equations

Accurate multistep multi-derivative collocation methods are derived for the numerical integration of chaotic systems. These methods have high order of accuracy, with A-stable regions of absolute stability. Although the calculation of the third derivative terms in the methods is relatively high compared to the rst and second derivative terms, the advantage gained makes them suitable for solving chaotic system of equations with large eigenvalues. The stability characteristic properties and order of accuracy of the methods are studied. Figurative comparisons of the solution curves obtained are in good agreement with the exact solutions which demonstrate practically the eectiveness of the proposed methods.

An efficient trigonometrical-fitted two-derivative multistep collocation (TF-TDMC) method using Legendre polynomials up to order five as the basis functions, has been developed for solving second-order ordinary differential equations with oscillatory solution effectively. Interpolation method of approximated power series and collocation technique of its second and third derivative are implemented in the construction of the methods. Two-derivative multistep collocation methods are developed in predictor and corrector form with varying collocation and interpolation points. Later, trigonometrically-fitting technique is implemented into TF-TDMC method, using the linear combination of trigonometrical functions, to produce frequency-dependent coefficients in TF-TDMC method. The stability of the TF-TDMC method, with fitted parameters, is thoroughly analyzed and has been proven to achieve zero stability. Stability polynomials and regions for predictor and corrector of TF-TDMC method are developed and plotted. In the operation of the TF-TDMC method, initial conditions and the frequency for each problem (based on the exact solutions) are identified. The frequency-dependent coefficients are then adjusted according to the identified frequency. Predictor and corrector steps are implemented to estimate and refine the values of the dependent variable and its derivative, ensuring that convergence is achieved.A numerical experiment demonstrates that the proposed method significantly outperforms other existing methods in the literature, achieving the lowest maximum global error with moderate computational time across all step sizes for solving second-order ordinary differential equations with oscillatory solutions. Additionally, it effectively addresses real-world perturbed Kepler problems. The results include a detailed discussion and analysis of the numerical performance.•An efficient two-derivative multistep collocation method in predictor-corrector mode with trigonometrically-fitting technique (TF-TDMC) is developed for direct solving second-order ordinary differential equations with oscillatory solution.•TF-TDMC method has been proved to acquire zero-stability and its stability region is analyzed.•TF-TDMC method is the best among all selected methods in solving second-order ordinary differential equations with oscillatory solution, including perturbed Kepler problem.

Generally, solutions to functional equations involving non-vanishing delays tend to exhibit lower regularity compared to those of smooth functions. In this context, we examine a first-order Volterra integro-differential equation (VIDE) with a non-vanishing delay, delving into the characteristics of its solutions. To enhance the accuracy of traditional one-step collocation methods [1], we employ multi-step collocation techniques to obtain numerical solutions for the VIDE with non-vanishing delay. The global convergence properties of the multi-step numerical approach are scrutinized using the Peano Kernel Theorem. Subsequently, for comparative analysis, we utilize a one-step collocation method to numerically solve this equation, showcasing the effectiveness and precision of the multi-step collocation method.

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

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.

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

Multistep collocation methods for Volterra integro-differential equations

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

A computational approach with the aid of the Linear Multistep Method (LMM) for the numerical solution of differential equations with initial value problems or boundary conditions has appeared several times in the literature due to its good accuracy and stability properties. The major objective of this article is to extend a multistep approach for the numerical solution of the Partial Differential Equation (PDE) originating from fluid mechanics in a two-dimensional space with initial and boundary conditions, as a result of the importance and utility of the models of partial differential equations in applications, particularly in physical phenomena, such as in convection-diffusion models, and fluid flow problems. Thus, a multistep collocation formula, which is based on orthogonal polynomials is proposed. Ninth-order Multistep Collocation Formulas (NMCFs) were formulated through the principle of interpolation and collocation processes. The theoretical analysis of the NMCFs reveals that they have algebraic order nine, are zero-stable, consistent, and, thus, convergent. The implementation strategies of the NMCFs are comprehensively discussed. Some numerical test problems were presented to evaluate the efficacy and applicability of the proposed formulas. Comparisons with other methods were also presented to demonstrate the new formulas’ productivity. Finally, figures were presented to illustrate the behavior of the numerical examples.

This paper presented a linear multistep method for solving fourth order initial value problems of ordinary differential equations. Collocation and interpolation methods are adopted in the derivation of the new numerical scheme which is further applied to finding direct solution of fourth order ordinary differentiation equations. This implementation strategy is more accurate and efficient than Adams–Bashforth Method solution. The newly derive scheme have better stabilities properties than that of the Adams-Bashforth Method. Numerical examples are included to illustrate the reliability and accuracy of the new methods.Keywords: Linear Multistep Methods, Region of Absolute Stability, Zero-Stability, Error Analysis, Collocation.

This paper examines the development of one step, five hybrid point method for the solution of first order initial value problems. We adopted the method of collocation and interpolation of power series approximate solution to generate a continuous linear multistep method. The continuous linear multistep method was evaluated at selected grid points to give the discrete linear multistep method. The method was implemented using a constant order predictor of order seven over an overlapping interval. The basic properties of the derived corrector was investigated and found to be zero stable, consistent and convergent. The region of absolute stability was also investigated. The method was tested on some numerical experiments and found to compare favorably with the existing methods.

This paper is devoted to studying the boundary value method for Volterra integral equations. High order numerical schemes are devised by using special multistep collocation methods, which depend on numerical approximations of the solution in the next several steps. Stability analysis illustrates these methods enjoy wide absolutely stable regions. With the help of efficient evaluation for highly oscillatory integrals, these methods are applied to solving Volterra integral equations with highly oscillatory kernels. Both theoretical and numerical results show they share the property that the higher the oscillation, the better the accuracy of the approximations.

We investigate multi-step Chebyshev spectral collocation method for Volterra integro-differential equations. We obtain numerical solution Y(t) and $$Y'(t)$$Yź(t) to approximate unknown function y(t) and its derivative $$y'(t)$$yź(t) while Y(t) and $$Y'(t)$$Yź(t) keep the relation that $$Y'(t)$$Yź(t) is the derivative of Y(t). We discuss existence and uniqueness of the solution to corresponding discrete system. We provide convergence analysis for proposed method. Numerical experiments are carried out to confirm theoretical results.