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A method is presented for calculating the Lie point symmetries of a scalar difference equation on a two-dimensional lattice. The symmetry transformations act on the equations and on the lattice. They take solutions into solutions and can be…
The paper develops the method for construction of families of particular solutions to some classes of nonlinear Partial Differential Equations (PDE). Method is based on the specific link between algebraic matrix equations and PDE.…
We discuss the use of inequalities to obtain the solution of certain variational problems on time scales.
Integrability of the differential constraints arising from the singularity analysis of two (1+1)-dimensional second-order evolution equations is studied. Two nonlinear ordinary differential equations are obtained in this way, which are…
This paper explores preconditioning the normal equation for non-symmetric square linear systems arising from PDE discretization, focusing on methods like CGNE and LSQR. The concept of ``normal'' preconditioning is introduced and a strategy…
Symmetry properties are at the basis of integrability. In recent years, it appeared that so called "twisted symmetries" are as effective as standard symmetries in many respects (integrating ODEs, finding special solutions to PDEs). Here we…
Here the polynomial interpolation approach is used to introduce the main results on multivariate normal algebraic systems. Next we bring a construction which shows that any standard algebraic system, with finite set of solutions, can be…
In this work, we consider solutions to (fully nonlinear) parabolic integro-differential equations with integrable interaction kernels. A typical equation would be that obtained by starting with, for $s\in(0,1)$, the $s$-fractional heat…
We investigate the integrability of Nonlinear Partial Differential Equations (NPDEs). The concepts are developed by firstly discussing the integrability of the KdV equation. We proceed by generalizing the ideas introduced for the KdV…
System of semilinear ordinary differential equation and fractional differential equation of distributed order is investigated and solved in a mild and classical sense. Such a system arises as a distributed derivative model of…
Nonlinear time fractional partial differential equations are widely used in modeling and simulations. In many applications, there are high contrast changes in media properties. For solving these problems, one often uses coarse spatial grid…
Modified scattering phenomena are encountered in the study of global properties for nonlinear dispersive partial differential equations in situations where the decay of solutions at infinity is borderline and scattering fails just barely.…
The method of separation of variables is significant, it has been applied to physics, engineering , chemistry and other fields. It allows to reduce the diffculity of problems by separating the variables from partial differential equation…
The notion of moment differentiation is extended to the set of generalized multisums of formal power series via an appropriate integral representation and accurate estimates of the moment derivatives. The main result is applied to…
In this paper we propose and solve a generalization of the Bernoulli Differential Equation, by means of a generalized fractional derivative. First we prove a generalization of Gronwall's inequality, which is useful for studying the…
We study a family of Li\'enard--type equations. Such equations are used for the description of various processes in physics, mechanics and biology and also appear as traveling--wave reductions of some nonlinear partial differential…
In this work we propose a mechanism for converting the spectral problem of vertex models transfer matrices into the solution of certain linear partial differential equations. This mechanism is illustrated for the…
We present an extension of a previously developed method employing the formalism of the fractional derivatives to solve new classes of integral equations. This method uses different forms of integral operators that generalizes the…
In this study, a new form of quadratic spline is obtained, where the coefficients are determined explicitly by variational methods. Convergence is studied and parity conservation is demonstrated. Finally, the method is applied to solve…
We consider difference equations with several non-monotone deviating arguments and nonnegative coefficients. The deviations (delays and advances) are, generally, unbounded. Sufficient oscillation conditions are obtained in an explicit…