scholarly journals Solution of Pantograph Equation by Collocation method using Orthogonal Exponential Polynomials(OEP)

2019 ◽  
Vol 1 (2) ◽  
pp. 126-127
Author(s):  
Muhammad Bilal ◽  
Norhayati Binit Rosli ◽  
Iftikar Ahmad ◽  
Mirza Rizwan Sajid
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Present Method ◽  
Numerical Technique ◽  
Exponential Polynomials ◽  
Pantograph Equation ◽  
Delay Differential

Novel matrix based numerical technique known as collocation method is implemented for the solution of pantograph differential equations (PDE) via truncated orthoexponential polynomial(OEP). To check applicability, reliability and efficiency of the methodology, here examine three examples of delay differential equations. At last the comparison made between proposed and reported methodologies and present method was perfect in agreement.

scholarly journals Numerical Solution of Pantograph-Type Delay Differential Equations Using Perturbation-Iteration Algorithms

2015 ◽  
Vol 2015 ◽  
pp. 1-10 ◽  
Author(s):  
M. Mustafa Bahşi ◽  
Mehmet Çevik
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Functional Differential Equations ◽  
Taylor Series Expansion ◽  
Perturbation Parameter ◽  
Iteration Algorithm ◽  
Proportional Delay ◽  
Pantograph Equation ◽  
Functional Differential ◽  
Delay Differential

The pantograph equation is a special type of functional differential equations with proportional delay. The present study introduces a compound technique incorporating the perturbation method with an iteration algorithm to solve numerically the delay differential equations of pantograph type. We put forward two types of algorithms, depending upon the order of derivatives in the Taylor series expansion. The crucial convenience of this method when compared with other perturbation methods is that this method does not require a small perturbation parameter. Furthermore, a relatively fast convergence of the iterations to the exact solutions and more accurate results can be achieved. Several illustrative examples are given to demonstrate the efficiency and reliability of the technique, even for nonlinear cases.

scholarly journals A new stable collocation method for solving a class of nonlinear fractional delay differential equations

2020 ◽  
Vol 85 (4) ◽  
pp. 1123-1153
Author(s):  
Lei Shi ◽  
Zhong Chen ◽  
Xiaohua Ding ◽  
Qiang Ma
Keyword(s):  
Differential Equations ◽  
Collocation Method ◽  
Delay Differential Equations ◽  
Linear Equations ◽  
Approximate Solutions ◽  
Orthonormal Bases ◽  
Fractional Delay ◽  
Nonlinear Condition ◽  
Delay Differential ◽  
Fractional Delay Differential Equations

AbstractIn this paper, a stable collocation method for solving the nonlinear fractional delay differential equations is proposed by constructing a new set of multiscale orthonormal bases of $W^{1}_{2,0}$ W 2 , 0 1 . Error estimations of approximate solutions are given and the highest convergence order can reach four in the sense of the norm of $W_{2,0}^{1}$ W 2 , 0 1 . To overcome the nonlinear condition, we make use of Newton’s method to transform the nonlinear equation into a sequence of linear equations. For the linear equations, a rigorous theory is given for obtaining their ε-approximate solutions by solving a system of equations or searching the minimum value. Stability analysis is also obtained. Some examples are discussed to illustrate the efficiency of the proposed method.

scholarly journals A Numerical Technique for Solving Nonlinear Singularly Perturbed Delay Differential Equations

2018 ◽  
Vol 23 (1) ◽  
pp. 64-78 ◽  
Author(s):  
A.S.V. Ravi Kanth ◽  
P. Murali Mohan Kumar
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Delay Differential Equations ◽  
Numerical Technique ◽  
Singularly Perturbed ◽  
Spline Method ◽  
Numerical Examples ◽  
Exponentially Fitted ◽  
Delay Differential ◽  
Quasilinearization Technique

This paper presents a numerical technique for solving nonlinear singu- larly perturbed delay differential equations. Quasilinearization technique is applied to convert the nonlinear singularly perturbed delay differential equation into a se- quence of linear singularly perturbed delay differential equations. An exponentially fitted spline method is presented for solving sequence of linear singularly perturbed delay differential equations. Error estimates of the method is discussed. Numerical examples are solved to show the applicability and efficiency of the proposed scheme.

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