This study is devoted to the numerical investigation of linear and nonlinear hyperbolic telegraph equation. We have proposed a wavelet collocation method based on Legendre polynomials for approximating the solution. Both the spatial and temporal variables, along with their derivatives, are approximated using the Legendre wavelet and its integration. The present approach is simple, consistent and straightforward. To assure the theoretical consistency of the method, an estimate for the upper bound of the error norm is provided. We have proved an exponential order of convergence which is better than the methods available in the literature. Some numerical experiments are carried out to justify the theoretical results and the outcomes confirm the computational efficiency of the proposed method.

A wavelet collocation method based on Hermite polynomials is proposed for the study of a class of 2D quasi-linear elliptic partial differential equations (PDEs) that arise frequently in applied mathematics, electromagnetic theory, nonlinear optics, weather forecasting, etc. The application of Hermite wavelets and their integration to 2D quasi-linear elliptic PDEs yielded a system of equations. For the theoretical aspect, the upper bound of the error norm is established to guarantee the convergence of the method. The proposed approach has an exponential rate of convergence, so it converges very rapidly. The proposed method can be uniformly adapted to investigate the solution of 2D singularly perturbed elliptic PDEs without modifying the current scheme. Some numerical simulations have been done to validate the theoretical findings. The maximum absolute errors are calculated for different numbers of collocation grids. The comparison of numerical findings with the existing methods concludes the superiority of the proposed method.

Abstract This paper investigates numerical solution of generalized space-time fractional Klein–Gordon equations (GSTFKGE) by using Gegenbauer wavelet method (GWM). The developed method makes use of fractional order integral operator (FOIO) for Gegenbauer wavelet, which is constructed by employing the definition of Riemann–Liouville fractional integral (RLFI) operator and Laplace transformation. The present algorithm is based on Gegenbauer wavelet jointly with FOIO to convert a GSTFKGE into a system of equations which is solved by using Newton’s technique. Additionally, the upper bound of error norm of the proposed method is calculated to validate the theoretical authenticity of the developed method. The comparison of numerical outcomes with the existing results in the literature and graphical illustrations show the accuracy and reliability of our method.

A wavelet collocation method based on Haar wavelet is proposed to investigate the numerical solution of time fractional diffusion equation (TFDE) on a metric star graph. We have utilized the Riemann–Liouville definition of fractional integral operator together with Haar wavelet to obtain the Riemann–Liouville fractional integral operator for Haar wavelet (RLFIO‐H). The application of RLFIO‐H and Haar wavelet to TFDE on metric star graph returns a system of linear algebraic equations. Solving this system yields wavelet coefficients and subsequently the solution. The convergence analysis of the proposed method is discussed to establish the theoretical authenticity of the method. The proposed method is tested on four benchmark problems to validate the theoretical findings. Moreover, the comparison of the developed method with the existing method shows the superiority and accuracy of the proposed method. To the best of the author's knowledge, the proposed work is one of the first collocation method in the literature that deals with the solution of TFDE on metric star graph using wavelet numerically.

A high resolution wavelet collocation method based on Gegenbauer polynomials is proposed for the solution of fourth-order time-fractional integro-differential equations (FTFIDE) with a weakly singular kernel. A Riemann-Liouville fractional integral operator for the Gegenbauer scaling function (RLFIO-G) is constructed using the definition of Riemann-Liouville (R-L) operator with the aid of Gegenbauer scaling function. The application of Gegenbauer scaling function and RLFIO-G to FTFIDE gives a system of linear algebraic equations, which can be quickly solved for unknown coefficients. With the aid of these coefficients, we get the approximate solution. We have also established the convergence analysis of the proposed method. We have tested the presented method on some numerical examples to demonstrate the accuracy. Moreover, we have compared the developed method with the existing method to conclude the superiority of the proposed method.

In this work, we have investigated p -type fractional neutral delay differential equations ( p -FNDDE) and p -type fractional neutral delay partial differential equations ( p -FNDPDE) via generalized Gegenbauer wavelet. Generalized Gegenbauer scaling function fractional integral operator (GGSFIO) is constructed using the Riemann–Liouville definition of fractional integral to handle the fractional derivatives present in p -FNDDE and p -FNDPDE. The operation of Gegenbauer wavelet basis and GGSFIO to p -FNDDE and p -FNDPDE returns a system of equations which is later solved by Newton’s method for unknown wavelet coefficients. With the help of these coefficients, we get the approximate solution. We have established the convergence analysis to assure the theoretical authenticity of the present method. The developed scheme is tested on several examples of p -FNDDE and p -FNDPDE to ensure computational convergence which validated the theoretical findings. The comparison of the numerical results of our method with the existing methods concludes the superiority of the proposed method.

A Legendre wavelet collocation method is proposed for solving a nonlinear coupled time fractional diffusion system. We have formulated a Riemann-Liouville fractional integral operator for Legendre wavelet (RLFIO-L) adopting the definition of Riemann-Liouville fractional integral operator combined with the Laplace transformation. Both the time and space variables are discretized in terms of the Legendre wavelet and RLFIO-L. The nonlinear coupled diffusion system is quasi-linearized by making use of the Newton's method. For theoretical concerns, the upper bound of error norm of the proposed method is estimated. Some numerical experiments are presented to authenticate the computational efficacy of the method.

This paper aims to develop an improved Hermite wavelet resolution method for solving space–time-fractional partial differential equations (STFPDE). Unlike the previous wavelet methods in which operational matrices are constructed by using orthogonal functions and block pulse functions, we have directly formulated the Riemann–Liouville fractional integral (RLFI) operator for Hermite wavelets of general order integration. We have also shown the error bounds of the established method to demonstrate the theoretical applicability of the proposed method. The accuracy of the developed method is tested via a descriptive comparison of the numerical results with those obtained from other existing methods. The investigative results validate that the introduced technique is stable, authentic, straightforward, and computationally reliable.

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.

In this paper, we introduce two different methods based on Gegenbauer wavelet and Bernoulli wavelet for the solution of neutral delay differential equations. These methods convert linear and nonlinear neutral delay differential equations into system of linear and nonlinear algebraic equations, respectively. After solving these equations, we get the approximate solutions. Here, we have used the Gegenbauer wavelet (for different values of μ) and Bernoulli wavelet and seen that both methods converge fast. We present six test problems consisting of five linear and one nonlinear, to illustrate the accuracy of present methods. Further, we compared our results with the results of existing methods present in the literature and have seen that our methods give more accurate results.

This paper is concerned with the Lane–Emden boundary value problems arising in many real-life problems. Here, we discuss two numerical schemes based on Jacobi and Bernoulli wavelets for the solution of the governing equation of electrohydrodynamic flow in a circular cylindrical conduit, nonlinear heat conduction model in the human head, and non-isothermal reaction–diffusion model equations in a spherical catalyst and a spherical biocatalyst. These methods convert each problem into a system of nonlinear algebraic equations, and on solving them by Newton’s method, we get the approximate analytical solution. We also provide the error bounds of our schemes. Furthermore, we also compare our results with the results in the literature. Numerical experiments show the accuracy and reliability of the proposed methods.

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