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In the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method.

The aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points.

An efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time.

The outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.

The purpose of this paper is to develop a new method based on operational matrices of Bernoulli wavelet for solving linear stochastic Itô-Volterra integral equations, numerically.

For this aim, Bernoulli polynomials and Bernoulli wavelet are introduced, and their properties are expressed. Then, the operational matrix and the stochastic operational matrix of integration based on Bernoulli wavelet are calculated for the first time.

By applying these matrices, the main problem would be transformed into a linear system of algebraic equations which can be solved by using a suitable numerical method. Also, a few results related to error estimate and convergence analysis of the proposed scheme are investigated.

Two numerical examples are included to demonstrate the accuracy and efficiency of the proposed method. All of the numerical calculation is performed on a personal computer by running some codes written in MATLAB software.

The purpose of this paper is to propose an efficient computational technique, which uses Haar wavelets collocation approach coupled with the Newton-Raphson method and solves the following class of system of Lane–Emden equations:

To deal with singularity, Haar wavelets are used, and to deal with the nonlinear system of equations that arise during computation, the Newton-Raphson method is used. The convergence of these methods is also established and the results are compared with existing techniques.

The authors propose three methods based on uniform Haar wavelets approximation coupled with the Newton-Raphson method. The authors obtain quadratic convergence for the Haar wavelets collocation method. Test problems are solved to validate various computational aspects of the Haar wavelets approach. The authors observe that with only a few spatial divisions the authors can obtain highly accurate solutions for both initial value problems and boundary value problems.

The results presented in this paper do not exist in the literature. The system of nonlinear singular differential equations is not easy to handle as they are singular, as well as nonlinear. To the best of the knowledge, these are the first results for a system of nonlinear singular differential equations, by using the Haar wavelets collocation approach coupled with the Newton-Raphson method. The results developed in this paper can be used to solve problems arising in different branches of science and engineering.

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.

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.

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.

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.