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We use "generalized" version of total variation, coarea formulas, isoperimetric inequalities to obtain sharp estimates for solutions (and for their gradients) to anisotropic elliptic equations with a lower order term, comparing them with…
We determine the most general group of equivalence transformations for a family of differential equations defined by an arbitrary vector field on a manifold. We also find all invariants and differential invariants for this group up to the…
We study admissible transformations and Lie symmetries for a class of variable-coefficient Burgers equations. We combine the advanced methods of splitting into normalized subclasses and of mappings between classes that are generated by…
In this paper, to begin with, we review six different analytical methods which are widely used to derive symmetries, integrating factors, multipliers, Darboux polynomials and integrals of second order nonlinear ordinary differential…
We show that any first order ordinary differential equation with a known Lie point symmetry group can be discretized into a difference scheme with the same symmetry group. In general, the lattices are not regular ones, but must be adapted…
A large class of initial-boundary value problems of linear evolution partial differential equations formulated on the half-line is analyzed via the unified transform method. In particular, explicit formulae are presented for the generalized…
We introduce a generalized finite difference method for solving a large range of fully nonlinear elliptic partial differential equations in three dimensions. Methods are based on Cartesian grids, augmented by additional points carefully…
We perform complete group classification of the general class of quasi linear wave equations in two variables. This class may be seen as a broad generalization of the nonlinear d'Alembert, Liouville, sin/sinh-Gordon and Tzitzeica equations.…
We carry out a detailed Lie point symmetry group classification of the Li\'enard type equation, $\ddot{x}+f(x)\dot{x}+g(x) = 0$, where $f(x)$ and $g(x)$ are arbitrary smooth functions of $x$. We divide our analysis into two parts. In the…
By using Lie symmetry methods, we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the Ermakov-Pinney equation. In this context, nonlinear…
Extensive work has been done on the group classification of systems of equations in the literature. This paper identifies the gap in the literature which concerns the group classification of systems of two autonomous nonlinear second-order…
We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence…
A complete geometric classification of symmetries of autonomous Hamiltonian mechanical systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results…
A general formalism to solve nonlinear differential equations is given. Solutions are found and reduced to those of second order nonlinear differential equations in one variable. The approach is uniformized in the geometry and solves…
Symplectic integration of autonomous Hamiltonian systems is a well-known field of study in geometric numerical integration, but for non-autonomous systems the situation is less clear, since symplectic structure requires an even number of…
This article reviews the use of differential forms and Lie derivatives to find symmetries of differential equations, as originally presented by Harrison and Estabrook, J. Math. Phys., 12 (1971), 653. An outline of the method is given,…
Within the framework of inverse Lie problem, we give some non-trivial examples of coupled Lie remarkable equations, \textit{i.e.}, classes of differential equations that are in correspondence with their Lie point symmetries. In particular,…
We show that any second order linear ordinary diffrential equation with constant coefficients (including the damped and undumped harmonic oscillator equation) admits an exact discretization, i.e., there exists a difference equation whose…
Fundamentals on Lie group methods and applications to differential equations are surveyed. Many examples are included to elucidate their extensive applicability for analytically solving both ordinary and partial differential equations.
We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle-Singer method. We describe a procedure to deduce all the integrals of motion associated with the given…