scholarly journals An Approximated Solutions for nth Order Linear Delay Integro-Differential Equations of Convolution Type Using B-Spline Functions and Weddle Method

2014 ◽  
Vol 11 (1) ◽  
pp. 166-177
Author(s):  
Baghdad Science Journal
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Convolution Type ◽  
Spline Functions ◽  
Exact Results ◽  
B Spline ◽  
Order Linear ◽  
Linear Delay

The paper is devoted to solve nth order linear delay integro-differential equations of convolution type (DIDE's-CT) using collocation method with the aid of B-spline functions. A new algorithm with the aid of Matlab language is derived to treat numerically three types (retarded, neutral and mixed) of nth order linear DIDE's-CT using B-spline functions and Weddle rule for calculating the required integrals for these equations. Comparison between approximated and exact results has been given in test examples with suitable graphing for every example for solving three types of linear DIDE's-CT of different orders for conciliated the accuracy of the results of the proposed method.

An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

2018 ◽  
Vol 28 (11) ◽  
pp. 2620-2649 ◽  
Author(s):  
Rajni Rohila ◽  
R.C. Mittal
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Solutions ◽  
Diffusion Equations ◽  
Spline Functions ◽  
Two Dimensional ◽  
Partial Differential ◽  
Content Type ◽  
B Spline ◽  
Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

A Numerical Approach for Solving High-Order Linear Delay Volterra Integro-Differential Equations

2018 ◽  
Vol 15 (05) ◽  
pp. 1850042 ◽  
Author(s):  
Şuayip Yüzbaşı ◽  
Murat Karaçayır
Keyword(s):  
Differential Equations ◽  
Approximate Solutions ◽  
Equation System ◽  
Numerical Approach ◽  
High Order ◽  
Inner Product ◽  
Present Scheme ◽  
Order Linear ◽  
Linear Delay ◽  
Residual Correction

In this study, a numerical method is proposed to solve high-order linear Volterra delay integro-differential equations. In this approach, we assume that the exact solution can be expressed as a power series, which we truncate after the [Formula: see text]-st term so that it becomes a polynomial of degree [Formula: see text]. Substituting the unknown function, its derivatives and the integrals by their matrix counterparts, we obtain a vector equivalent of the equation in question. Applying inner product to this vector with a set of monomials, we are left with a linear algebraic equation system of [Formula: see text] unknowns. The approximate solution of the problem is then computed from the solution of the resulting linear system. In addition, the technique of residual correction, whose aim is to increase the accuracy of the approximate solutions by estimating the error of those solutions, is discussed briefly. Both the method and this technique are illustrated with several examples. Finally, comparison of the present scheme with other methods is made wherever possible.

Oscillations of second order linear delay differential equations

1988 ◽  
Vol 27 (1-3) ◽  
pp. 109-123 ◽  
Author(s):  
M.R.S. Kulenvić ◽  
G. Ladas ◽  
Y.G. Sficas
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Second Order ◽  
Order Linear ◽  
Linear Delay ◽  
Delay Differential
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