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Perturbation Methods In Applied Mathematics

Collocation methods based on Gegenbauer and Bernoulli wavelets for solving neutral delay differential equations

Herein, we developed and analyzed a new fractal–fractional (FF) operational matrix for orthonormal normalized ultraspherical polynomials. We used this matrix to handle the FF Riccati differential equation with the new generalized Caputo FF derivative. Based on the developed operational matrix and the spectral Tau method, the nonlinear differential problem was reduced to a system of algebraic equations in the unknown expansion coefficients. Accordingly, the resulting system was solved by Newton’s solver with a small initial guess. The efficiency, accuracy, and applicability of the developed numerical method were checked by exhibiting various test problems. The obtained results were also compared with other recent methods, based on the available literature.

Orthonormal Ultraspherical Operational Matrix Algorithm for Fractal–Fractional Riccati Equation with Generalized Caputo Derivative

A new third order convergent numerical solver for continuous dynamical systems

In this paper we proposed wavelet based collocation methods for solving neutral delay differential equations. We use Legendre wavelet, Hermite wavelet, Chebyshev wavelet and Laguerre wavelet to solve the neutral delay differential equations numerically. We solve five linear and one nonlinear problem to demonstrate the accuracy of wavelet series solution. Wavelet series solution converges fast and gives more accurate results in comparison to other methods present in literature. We compare our results with Runge-Kutta-type methods by Wang et al. [1] and one-leg θ methods by Wang et al. [2] and observe that our results are more accurate.

Wavelet collocation methods for solving neutral delay differential equations

Haar wavelet series solution for solving neutral delay differential equations

\nPurpose\nThe 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.\n\n\nDesign/methodology/approach\nBecause 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.\n\n\nFindings\nThis 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.\n\n\nOriginality/value\nThis 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.\n

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

In this paper, we apply Haar wavelet collocation method to solve the linear and nonlinear second-order singularly perturbed differential difference equations and singularly perturbed convection delayed dominated diffusion equations, arising in various modeling of chemical processes. First, we transform delay term by using Taylor expansion and then apply Haar wavelet method. To show the robustness, accuracy and efficiency of the method, three problems of second-order singularly perturbed differential difference equations and three problems of convection delayed dominated diffusion equations have been solved. Also, results are compared with the exact solution of the problems and methods existing in the literature, which confirms the superiority of the Haar wavelet collocation method. We obtained accurate numerical solution of problems by increasing the level of resolutions.

Solution of singularly perturbed differential difference equations and convection delayed dominated diffusion equations using Haar wavelet

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