Related papers: Convolution type integral equations in the conserv…
Finite difference schemes that preserve two conservation laws of a given partial differential equation can be found directly by a recently-developed symbolic approach. Until now, this has been used only for equations with quadratic…
A class of variable coefficient (1+1)-dimensional nonlinear reaction-diffusion equations of the general form $f(x)u_t=(g(x)u^nu_x)_x+h(x)u^m$ is investigated. Different kinds of equivalence groups are constructed including ones with…
We study a class of degenerate convection diffusion equations with a fractional nonlinear diffusion term. These equations are natural generalizations of anomalous diffusion equations, fractional conservations laws, local convection…
We propose a new integral based on Taylor measures, study its properties extensively, and we illustrate that it includes many concepts from mathematics as special cases. In particular, the new integral emerges as a generalization of the…
In this work we revise the most recent developments concerning the study of first order problems regarding differential equations with involutions. We take into account two cases: the case of initial conditions and constant coefficients and…
Fifth order, quasi-linear, non-constant separant evolution equations are of the form u_t=A\frac{\partial^5 u}{\partial x^5}+\tilde{B}, where A and \tilde{B} are functions of x, t, u and of the derivatives of u with respect to x up to order…
We study the steady states and dynamics of a thin film-type equation with non-conserved mass in one dimension. The evolution equation is a nonlinear fourth-order degenerate parabolic PDE motivated by a model of volatile viscous fluid films…
We study a class of evolutionary partial differential systems with two components related to second order (in time) non-evolutionary equations of odd order in spatial variable. We develop the formal diagonalisation method in symbolic…
A general procedure for constructing conservative numerical integrators for time dependent partial differential equations is presented. In particular, linearly implicit methods preserving a time discretised version of the invariant is…
We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary…
Pattern formation in systems with a conserved quantity is considered by studying the appropriate amplitude equations. The conservation law leads to a large-scale neutral mode that must be included in the asymptotic analysis for pattern…
An analytical approach to convolution of functions, which appear in perturbative calculations, is discussed. An extended list of integrals is presented.
This paper present a numerical method for solving nonlinear Fredholm integral equations. The method is based upon Newton type approximations. Illustrative examples are included to demonstrate the validity and applicability of the technique.
The factorisation method commonly used in linear supersymmetric quantum mechanics is extended, such that it can be applied to nonlinear quantum mechanical systems. The new method is distinguishable from the linear formalism, as the…
Recently, a new fractional derivative called the conformable fractional derivative is given which is based on the basic limit definition of the derivative in [1]. Then, the fractional versions of chain rules, exponential functions,…
In this paper, we develop a new deformation and generalization of the Natural integral transform based on the conformable fractional $q$-derivative. We obtain transformation of some deformed functions and apply the transform for solving…
The invariant subspace method is refined to present more unity and more diversity of exact solutions to evolution equations. The key idea is to take subspaces of solutions to linear ordinary differential equations as invariant subspaces…
In this paper we study the generalized variable-coefficient Gardner equations of the form $u_t + A(t)u^n\,u_x+ C(t)\,u^{2n}u_x + B(t)\,u_{xxx} + Q(t)\,u =0$. This class broadens out many other equations previously considered: Johnpillai and…
An effective algorithmic method is presented for finding the local conservation laws for partial differential equations with any number of independent and dependent variables. The method does not require the use or existence of a…
In this paper, we apply the binary Bell polynomial approach to a (2+1) dimensional nonlinear evolution equation. Namely, this study is an integrability work. Bilinear formalism, bilinear Backlund transformation, Lax pair of referred…