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In this paper, we present an approximate solution for solving the nonlinear Rosenau–Hyman equation. The method is based on adapting the wavelet technique accompanied with the Hermit polynomials. Convergence analysis for the proposed method is being investigated, proving that the Hermite wavelet expansion is uniformly convergent. A test example is presented for different values of the parameters, and the obtained results are compared to other relevant methods from the literature. The process proves to have the ability to produce accurate results than the other compared methods. Some graphical representations for the problem are drawn to illustrate the behavior of the solution.

In this article, a new method is generated to solve nonlinear Lane–Emden type equations using Legendre, Hermite and Laguerre wavelets. We are interested to note that these wavelets will give same solutions with good accuracy. Theorems on convergence analysis are stated and proved on the spaces, which are created by Legendre, Hermite and Laguerre wavelets bases and justified these spaces are equivalent to polynomial linear space generated by general polynomial basis. The main idea for obtaining numerical solutions depends on converting the differential equation with initial and boundary conditions into a system of linear or nonlinear algebraic equations with unknown coefficients. A very high level of accuracy reflects the reliability of this scheme for such problems.

Heat transfer problems are of critical importance in almost all areas of engineering and technology, especially in mechanical engineering. Over several decades, engineers and scientists have been working on solving these problems by designing and optimizing components and systems needed to transfer, handle, and store thermal energy. Heat can be transmitted via different mechanisms, like conduction and convection. In this article, two nonlinear heat transfer problems are solved by allowing for the changeable specific heat coefficient. The mathematical calculations are permitted out via a Hermite wavelet method (HWM). By using HWM, the nonlinear unnatural governing equations are abridged to an interconnected system of algebraic equations. The current outcomes are compared with the Homotopy perturbation method (HPM), variational iteration method (VIM), differential transformation method (DTM), and the accurate solutions to confirm the exactness of the future method. The HWM method is helpful and practical for solving the nonlinear heat diffusion equations linked through variable thermal conductivity conditions. Moreover, the equation outcomes in boundary points, and scheming the inaccuracy differences with the accurate amount, the advantages and disadvantages of this technique are considered. The results reveal that the HWM can attain more appropriate consequences in predicting the solution of such problems.