A New Multi-Step Method for Solving Delay Differential Equations using Lagrange Interpolation

Abstract

This paper presents 2-step p-th order (p = 2,3,4) multi-step methods that are based on the combination of both polynomial and exponential functions for the solution of Delay Differential Equations (DDEs). Furthermore, the delay argument is approximated using the Lagrange interpolation. The local truncation errors and stability polynomials for each order are derived. The Local Grid Search Algorithm (LGSA) is used to determine the stability regions of the method. Moreover, applicability and suitability of the method have been demonstrated by some numerical examples of DDEs with constant delay, time dependent and state dependent delays. The numerical results are compared with the theoretical solution as well as the existing Rational Multi-step Method2 (RMM2).

A Higher-order Block Method for Numerical Approximation of Third-order Boundary Value Problems in ODEs

“…The matrix parameter A (0) , A (1) , C (0) , C (3) , D (0) , D (4) , H (0) , H (1) are the square matrices whose arrays are the coefficients (10) and are defined as below.…”

“…The HBM is implemented in block method together with the aid of Newton-Raphson approach via a Mathematica 11.0 code which uses f-solve for linear and findroot for non-linear to simultaneously generate the solution at the initial point to the terminal point while adjusting for boundary conditions. Meanwhile, each block integrators in (10), (11) and (12) forms a system of equations which is applied along with the Newton's method. The starting values in the application of the Newton's Raphson method which are considered as the approximations provided by the Taylor series expansion formulas…”

“…Consequently, the solution of ( 1) is simultaneously generate on the entire interval of integration by using the main method (10) for n = 0, 1, ..., N 2 to obtain 3N − 2 equations and the additional method (11) and (12) which are employed to complete the set of equations needed to simultaneously solve 3N by 3N system of equations for solving (1) directly.…”

“…On the other hand, the shooting method suffers inaccuracy and instability while the finite different method is very demanding and does not give a satisfactory results. Other numerical methods for solving ordinary differential equations also exist in literature [5][6][7][8][9][10][11][12][13][14][15].…”

An Accuracy-preserving Block Hybrid Algorithm for the Integration of Second-order Physical Systems with Oscillatory Solutions

“…The basic properties of the method were analysed and the method was implemented on some linear and nonlinear second order differential equations. Other researchers that also developed direct methods for solving problems of the form (1) are [12][13][14][15][16][17][18][19][20][21][22][23].…”