Haar wavelet series solution for solving neutral delay differential equations

2021

Tamkang J. Math.

Self Cite

An efficient Haar wavelet collocation method is proposed for the numerical solution of singularly perturbed differential equations with delay and shift. Taylor series (upto the first order) is used to convert the problem with delay and shift into a new problem without the delay and shift and then solved by Haar wavelet collocation method, which reduces the time and complexity of the system. Further, we apply the Haar wavelet collocation method directly to solve the problems. Also, we demonstrated several test examples to show the accuracy and efficiency of the Haar wavelet collocation method and compared our results with the finite difference and fitted operator finite difference method [11], [28].

Section: Resultsmentioning

confidence: 99%

“…Due to lack of differentiability of Haar wavelets, authors move towards integration approach instead of differentiation [3,9,21,23,26,28,30].…”

mentioning

confidence: 99%

Treatment of Singularly Perturbed Differential Equations with Delay and Shift Using Haar Wavelet Collocation Method

Raza

2021

Tamkang J. Math.

Self Cite

“…Oruç (2018a) developed a computational method based on Hermite wavelet for the solution of two-dimensional Sobolev and regularized long wave equations in fluids. Raza and Khan (2019) have dealt with the numerical solution of neutral delay differential equation (NDDE) by applying Haar wavelet. Faheem et al (2020) investigated NDDE via Gegenbauer and Bernoulli wavelets, and Zogheib et al (2017) used computational method based on Bernoulli wavelet for solving diffusion and wave equation.…”

Section: Introductionmentioning

confidence: 99%

Solution of third-order Emden–Fowler-type equations using wavelet methods

Faheem

Raza

2021

Self Cite

Purpose The numerical solution of third-order boundary value problems (BVPs) has a great importance because of their applications in fluid dynamics, aerodynamics, astrophysics, nuclear reactions, rocket science etc. The purpose of this paper is to develop two computational methods based on Hermite wavelet and Bernoulli wavelet for the solution of third-order initial/BVPs. Design/methodology/approach Because of the presence of singularity and the strong nonlinear nature, most of third-order BVPs do not occupy exact solution. Therefore, numerical techniques play an important role for the solution of such type of third-order BVPs. The proposed methods convert third-order BVPs into a system of algebraic equations, and on solving them, approximate solution is obtained. Finally, the numerical simulation has been done to validate the reliability and accuracy of developed methods. Findings This paper discussed the solution of linear, nonlinear, nonlinear singular (Emden–Fowler type) and self-adjoint singularly perturbed singular (generalized Emden–Fowler type) third-order BVPs using wavelets. A comparison of the results of proposed methods with the results of existing methods has been given. The proposed methods give the accuracy up to 19 decimal places as the resolution level is increased. Originality/value This paper is one of the first in the literature that investigates the solution of third-order Emden–Fowler-type equations using Bernoulli and Hermite wavelets. This paper also discusses the error bounds of the proposed methods for the stability of approximate solutions.

“…Maleknejad et al [12] developed the numerical method for the solution Volterra IEs of first, second, and singular type of equations by the use of Bernstein approximation. Raza and Khan [13] found solution of neutral delay differential equations. Ghasemia and Kajani [14] utilized Chebyshev wavelets to find the solution time delay systems.…”

Section: Introductionmentioning

confidence: 99%

Solution of the Systems of Delay Integral Equations in Heterogeneous Data Communication through Haar Wavelet Collocation Approach

Amin

et al. 2021

Complexity

In this work, the Haar collocation scheme is used for the solution of the class of system of delay integral equations for heterogeneous data communication. The Haar functions are considered for the approximation of unknown function. By substituting collocation points and applying the Haar collocation technique to system of delay integral equations, we have obtained a linear system of equations. For the solution of this system, an algorithm is developed in MATLAB software. The method of Gauss elimination is utilized for the solution of this system. Finally, by using these coefficients, the solution at collocation points is obtained. The convergence of Haar technique is checked on some test problems.