Related papers: Generalized Integrals and Solvability
We present a slight generalization of the notion of completely integrable systems to get them being integrable by quadratures. We use this generalization to integrate dynamical systems on double Lie groups.
We use some general properties, presented in previous work, to evaluate special cases of integrals relating Rogers-Ramanujan continued fraction, eta function and elliptic integrals.
We discuss alternative iteration methods for differential equations. We provide a convergence proof for exactly solvable examples and show more convenient formulas for nontrivial problems.
The problem of discretization of Darboux integrable equations is considered. Given a Darboux integrable continuous equation, one can obtain a Darboux integrable differential-discrete equation, using the integrals of the continuous equation.…
We develop the concept of pluri-Lagrangian structures for integrable hierarchies. This is a continuous counterpart of the pluri-Lagrangian (or Lagrangian multiform) theory of integrable lattice systems. We derive the multi-time Euler…
In this paper, we analyze nonlinear differential equations subject to generalized boundary conditions. More specifically, we provide a framework from which we can provide conditions, which are straightforward to check, for the solvability…
Several integrability problems of differential equations are addressed by using the concept of $\mathcal{C}^{\infty}$-structure, a recent generalization of the notion of solvable structure. Specifically, the integration procedure associated…
A class of second-order differential equations commonly arising in physics applications are considered, and their explicit hypergeometric solutions are provided. Further, the relationship with the Generalized and Universal Associated…
A systematic study of the discrete second order projective system is presented, complemented by the integrability analysis of the associated multilinear mapping. Moreover, we show how we can obtain third order integrable equations as the…
In this article we present logarithmic methods for solving first order and second order ordinary differential equations. The essence of the method is that we apply the basic properties derivatives and logarithms to reduce the number of…
The great innovation of the Generalized Theorem is that it gives us the philosophy to work out the knowledge that the number of roots of an equation depends on the subfields of the functional terms of the equation they generate. Thus, the…
In this paper, the conformable Laguerre and associated Laguerre differential equations are solved using the Laplace transform. The solution is found to be in exact agreement with that obtained using the power series. In addition some of…
We discuss an elementary derivation of variational symmetries and corresponding integrals of motion for the Lagrangian systems depending on acceleration. Providing several examples, we make the manuscript accessible to a wide range of…
We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete…
Line integration of generalized functions is studied. Second order partial differential equations with piecewise continuous and generalized variable coefficients over Cayley-Dickson algebras are investigated. Formulas for integrations of…
The method of exhaustion is generalized to a simple formula that can be used to integrate functions under very general conditions, provided that the integral exists. Both a geometric proof (following the usual procedure for the method of…
The sets of the integrable lattice equations, which generalize the Toda lattice, are considered. The hierarchies of the first integrals and infinitesimal symmetries are found. The properties of the multi-soliton solutions are discussed.
We propose new linear combinations of compositions of a basic second-order scheme with appropriately chosen coefficients to construct higher order numerical integrators for differential equations. They can be considered as a generalization…
We evaluate several arctangent and logarithmic integrals depending on a parameter. This provides a closed form summation of certain series and also gives integral and series representation of some classical constants.
We consider an integrable generalization of the sine-Gordon (sG) equation that was earlier derived by one of the authors using bi-Hamiltonian methods. This equation is related to the sG equation in the same way that the Camassa-Holm…