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In this research, Genocchi wavelets method, a quite new type of wavelet-like basis, is adopted to obtain a numerical solution for the classical and time-fractional Rosenau-Hyman or K(n, n) equation arising in the formation of patterns in liquid drops. The considered partial differential equation can be transformed into a system of non-linear algebraic equations by utilizing the wavelets method including an integral operational matrix and then discretizing the equation at the collocation points. The system can be simply solved by several traditional methods. Finally, the algorithm is implemented for some numerical examples and the numerical solutions are compared with the exact solutions using MAPLE. The obtained results are demonstrated using figures and tables. When the results are compared, it is evinced that the algorithm is quite effective and advantageous due to its easily computable algorithm, high accuracy, and less process time. (C) 2021 THE AUTHORS. Published by Elsevier BV on behalf of Faculty of Engineering, Alexandria University.

In this study, the Jacobi wavelet collocation method is studied to derive a solution of the time-fractional Fisher?s equation in Caputo sense. Jacobi wavelets can be considered as a generalization of the wavelets since the Gegenbauer, and thus also Chebyshev and Legendre polynomials are a special type of the Jacobi polynomials. So, more accurate and fast convergence solutions can be possible for some kind of problems thanks to Jacobi wavelets. After applying the proposed method to the considered equation and discretizing the equation at the collocation points, an algebraic equation system is derived and solving the equation system is quite sim?ple rather than solving a non-linear PDE. The obtained values of our method are checked against the other numerical and analytic solution of considered equation in the literature and the results are visualized by using graphics and tables so as to reveal whether the method is effectiveness or not. The obtained results evince that the wavelet method is quite proper because of its simple algorithm, high accuracy and less CPU time for solving the considered equation.

In this research, our main motivation is to find the novel analytical solutions of $$(2+1)$$ dimensional Heisenberg ferromagnetic spin equation, which describes the nonlinear dynamics of the ferromagnetic materials by using the extended rational $$sine-cosine$$ and $$sinh-cosh$$ methods. The considered PDE is converted to an ODE by applying a wave transformation, and then the solutions of the ODE are supposed to be in the rational forms of trigonometric functions. After substituting the solutions to the ODE and doing some basic calculations, a system of algebraic equations is derived. So, finding the solutions of the PDE turns into a problem of solving an algebraic system of equations. The unknown coefficients in the solutions that are in the rational form are found by solving the obtained system. The methods are powerful and can be applied to find exact solutions to lots of PDEs in mathematical physics.