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Existence, regularity and location of solutions to quasilinear singular elliptic systems with general gradient dependence are established developing a method of sub-supersolution. The abstract theorems involving sub-supersolutions are…
We use the framework of Colombeau algebras of generalized functions to study existence and uniqueness of global generalized solutions to mixed non-local problems for a semilinear hyperbolic system. Coefficients of the system as well as…
We introduce a discrete scheme for second order fully nonlinear parabolic PDEs with Caputo's time fractional derivatives. We prove the convergence of the scheme in the framework of the theory of viscosity solutions. The discrete scheme can…
We recommended consequent discrete combinatorial research in mathematical physics. Here we show an example how discretization of partial differential equations can be done and that quickly unexpected new findings can result from research in…
To approximate solutions of complex nonlinear partial differential equations remains a computational challenge, especially for sets of equations relevant in industry, such as Euler or Navier-Stokes equations. Even the most sophisticated…
We introduce and analyze a new mixed finite element method with reduced symmetry for the standard linear model in viscoelasticity. Following a previous approach employed for linear elastodynamics, the present problem is formulated as a…
We investigate the existence and the singular structure of delta wave solutions to a semilinear strictly hyperbolic equation with strongly singular initial and boundary conditions. The boundary conditions are given in nonlocal form with a…
We explicate a procedure to solve general linear differential equations, which connects the desired solutions to monomials x^m of an appropriate degree m. In the process the underlying symmetry of the equations under study, as well as that…
In this paper, we establish a relation between two seemingly unrelated concepts for solving first-order hyperbolic quasilinear systems of partial differential equations in many dimensions. These concepts are based on a variant of the…
We propose and analyze a structure-preserving space-time variational discretization method for the Cahn-Hilliard-Navier-Stokes system. Uniqueness and stability for the discrete problem is established in the presence of concentration…
In this paper, we describe a semi-discrete method for a numerical resolution of a type of partial differential equations, called the method of lines (MOL). This method is based on the discretization of all but one of the variables of the…
We consider a system of differential equations and obtain its solutions with exponential asymptotics and analyticity with respect to the spectral parameter. Solutions of such type have importance in studying spectral properties of…
The objective of this paper is to investigate the existence and the forms of the pair of finite order entire and meromorphic solutions of some certain systems of Fermat-type partial differential-difference equations of several complex…
In this article, we present a numerical approach to ensure the preservation of physical bounds on the solutions to linear and nonlinear hyperbolic convection-reaction problems at the discrete level. We provide a rigorous framework for error…
We present a robust computational framework for the numerical solution of a hyperbolic 6-equation single-velocity two-phase system. The system's main interest is that, when combined with instantaneous mechanical relaxation, it recovers the…
Starting from a non-local version of the Prigogine-Herman traffic model, we derive a natural hierarchy of kinetic discrete velocity models for traffic flow consisting of systems of quasi-linear hyperbolic equations with relaxation terms.…
We study solution techniques for parabolic equations with fractional diffusion and Caputo fractional time derivative, the latter being discretized and analyzed in a general Hilbert space setting. The spatial fractional diffusion is realized…
We construct so called Darboux matrices and fundamental solutions in the important case of the generalised Hamiltonian (or canonical) systems depending rationally on the spectral parameter. A wide class of explicit solutions is obtained in…
The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion of a subclass of these methods, we derive conditions for arbitrarily high order. We derive…
We propose a method by which to examine all possible partial difference Lax pairs that consist of 'two by two' discrete linear problems, where the matrices contain one separable term in each entry. We thereby derive new, higher-order…