We exhibit a new decomposition of the nonlinearity for Muskat equation and use it to commute Fourier multipliers with equation. This allows study solutions critical regularity. As corollary, we obtain first well-posedness result arbitrary large data in space H?2(R2)?W1,?(R2). Moreover, prove existence initial which are not Lipschitz.

The ultimate goal of this study is to develop a numerically effective approximation technique acquire numerical solutions the integer and fractional-order Bratu singular Lane–Emden-type problems especially with exponential nonlinearity. Both initial boundary conditions were considered fractional derivative being in Liouville–Caputo sense. In direct approach, generalized Bessel matrix method bas...

In this paper, an efficient numerical scheme based on uniform Haar wavelets is used to solve the non-planar Burgers equation. The quasilinearization technique is used to conveniently handle the nonlinear terms in the non-planar Burgers equation. The basic idea of Haar wavelet collocation method is to convert the partial differential equation into a system of algebraic equations that involves a ...