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A new iterative technique is presented for solving of initial value problem for certain classes of multidimensional linear and nonlinear partial differential equations. Proposed iterative scheme does not require any discretization,…
In this work we introduce the notion of differential-algebraic ansatz for the heat equation and explicitly construct heat equation and Burgers equation solutions given a solution of a homogeneous non-linear ordinary differential equation of…
Classification theory on the existence and non-existence of local in time solutions for initial value problems of nonlinear heat equations are investigated. Without assuming a concrete growth rate on a nonlinear term, we reveal the…
We study a second-order parabolic equation with divergence form elliptic operator, having piecewise constant diffusion coefficients with two points of discontinuity. Such partial differential equations appear in the modelization of…
We present a family of integral equation-based solvers for the linear or semilinear heat equation in complicated moving (or stationary) geometries. This approach has significant advantages over more standard finite element or finite…
Computing solutions to partial differential equations using the fast Fourier transform can lead to unwanted oscillatory behavior. Due to the periodic nature of the discrete Fourier transform, waves that leave the computational domain on one…
In this paper we provide a new (probabilistic) proof of a classical result in partial differential equations, viz. if $\phi$ is a tempered distribution, then the solution of the heat equation for the Laplacian, with initial condition…
Delay differential equations are of great importance in science, engineering, medicine and biological models. These type of models include time delay phenomena which is helpful for characterising the real-world applications in machine…
Using matrix identities, we construct explicit pseudo-exponential-type solutions of linear Dirac, Loewner and Schr\"odinger equations depending on two variables and of nonlinear wave equations depending on three variables.
This paper shows how the enclosure method which was originally introduced for elliptic equations can be applied to inverse initial boundary value problems for parabolic equations. For the purpose a prototype of inverse initial boundary…
This paper is divided into three parts. The first part focuses on periodic layer heat potentials, demonstrating their smooth dependence on regular perturbations of the support of integration. In the second part, we present an application of…
A condition which guaranties the exponential decay of the solutions of the initial-boundary value problem for the damped wave equation is proved. A method for the effective computability of the coefficient of exponential decay is also…
We show the continuous dependence of solutions of linear nonautonomous second order parabolic partial differential equations (PDEs) with bounded delay on coefficients and delay. The assumptions are very weak: only convergence in the weak-*…
In this paper, we study well-posedness and exponential stability for semilinear second order evolution equations with memory and time-varying delay feedback. The time delay function is assumed to be continuous and bounded. Under a suitable…
In this paper, we analyze a semilinear damped second order evolution equation with time-dependent time delay and time-dependent delay feedback coefficient. The nonlinear term satisfies a local Lipschitz continuity assumption. Under…
Functional evolution equations are used in the modeling of numerous physical processes. In this work, our main tool is perturbation theory of strongly continuous semigroups. The advantage of this technique is that one can provide functional…
We consider the Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. We establish local well-posedness result in Orlicz spaces. We derive the existence of global solutions for small initial data. We…
This paper develops an explicit spectral representation for solutions of a one-dimensional linear wave equation with a constant time delay. The model is considered on a bounded interval with non-homogeneous Dirichlet boundary data and a…
Variational quantum algorithms are a promising tool for solving partial differential equations. The standard approach for its numerical solution are finite difference schemes, which can be reduced to the linear algebra problem. We consider…
We suggest a modification of the operator exponential method for the numerical solving the difference linear initial boundary value problems. The scheme is based on the representation of the difference operator for given boundary conditions…