The quasi-classical ∂̄-dressing method is used to derive compact generating equations for dispersionless hierarchies. Dispersionless Kadomtsev-Petviashvili (KP) and two-dimensional Toda lattice (2DTL) hierarchies are considered as illustrative examples.

The exact solutions of some conformable time fractional PDEs are presented explicitly. The modified Kudryashov method is applied to construct the solutions to the conformable time fractional Regularized Long WaveBurgers (RLW-Burgers, potential Korteweg-de Vries (KdV) and clannish random walker’s parabolic (CRWP) equations. Initially, the predicted solution in the finite series of a rational for...

The computational complexity of one-dimensional time fractional reaction-diffusion equation is O(N²M) compared with O(NM) for classical integer reaction-diffusion equation. Parallel computing is used to overcome this challenge. Domain decomposition method (DDM) embodies large potential for parallelization of the numerical solution for fractional equations and serves as a basis for distributed, ...

In this paper we observe that the existence of an sl(2,R) can help us in organizing the analysis and computation of time-dependent symmetries of nonlinear evolution equations. We apply this idea to the Burgers and Ibragimov–Shabat equations. We then use sl(2,R) to compute their Bäcklund transformations. This leads to unexpected consequences for the Kadomtsev–Petviashvili equation: we show that ...

By space-fractional (or L evy-Feller) diiusion processes we mean the processes governed by a generalized diiusion equation which generates all L evy stable probability distributions with index (0 < 2), including the two symmetric most popular laws, Cauchy (= 1) and Gauss (= 2). This generalized equation is obtained from the standard linear diiusion equation by replacing the second-order space d...

Here u = u(x, y, t) is a scalar function, x and y are respectively the longitudinal and transverse spatial coordinates, subscripts x, y, t denote partial derivatives, and σ2 = ±1. The case σ = 1 is known as the KPII equation, and models, for instance, water waves with small surface tension. The case σ = i is known as the KPI equation, and may be used to model waves in thin films with high surfa...

By using the complete discrimination system for the polynomial method, the classification of single traveling wave solutions to the generalized Kadomtsev-Petviashvili equation without dissipation terms in is obtained. 2 p 