Wavelets have become a powerful tool for having applications in almost all the areas of engineering and science such as numerical simulation of partial differential equations. In this paper, we present the Haar wavelet method (HWM) to solve the linear and nonlinear Klein–Gordon equations which occur in several applied physics fields such as, quantum field theory, fluid dynamics, etc. The fundamental idea of HWM is to convert the Klein–Gordon equations into a group of algebraic equations, which involve a finite number of variables. The examples are given to demonstrate the numerical results obtained by HWM, are compared with already existing numerical method i.e. finite difference method (FDM) and exact solution to confirm the good accuracy of the presented scheme.

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In this paper, we applied the adaptive grid Haar wavelet collocation method (AGH-WCM) for the numerical solution of parabolic partial differential equations (PDEs). The approach of AGHWCM for the numerical solution of parabolic PDEs is mentioned, the obtained numerical results, error analysis are presented in figures and tables. This shows that, the AGHWCM gives better accuracy than the HWCM and FDM. Some of the test problems are taken for demonstrating the validity and applicability of the AGHWCM.

Recently, wavelet-based numerical methods have been newly developed in the areas of science and engineering. In this paper, we proposed a full-approximation scheme for the numerical solution of Burgers’ equation using biorthogonal wavelet filter coefficients as prolongation and restriction operators. The presented scheme gives higher accuracy in terms of higher convergence in less CPU time, which has been illustrated through the test problem.

This paper presents biorthogonal wavelet-based multigrid (BWMG) and full approximation scheme (FAS) for the numerical solution of parabolic partial differential equations (PPDEs), which are working horse behind many commercial applications like finger print image compression. Performance of the proposed schemes is better than the existing ones in terms of super convergence with less computational time. Some of the test problems are taken to demonstrate the applicability and validity of the method.