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It is well known that the real and imaginary parts of any holomorphic function are harmonic functions of two variables. In this paper we generalize this property to finite-dimensional commutative algebras. We prove that if some basis of a…
In this paper, we propose a procedure for constructing an infinite number of families of solutions of given linear differential equations with partial derivatives with constant coefficients. We use monogenic functions that are defined on…
We describe the solutions to a family of rotationally symmetric second order partial differential equations in the complex plane that arises from a four-dimensional complex Lie algebra whose spanning set generates the algebra from which…
We establish the existence of positive solutions for a system of coupled fourth-order partial differential equations on a bounded domain $\Omega \subset \mathbb{R}^n$\begin{align*} \left\{\begin{array}{l} \Delta^2u_1 +\beta_1 \Delta…
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…
This book encompasses both traditional and modern methods treating partial differential equation (PDE) of first order and second order. There is a balance in making a selfcontained mathematical text and introducing new subjects. The Lie…
We are concerned with the arithmetic of solutions to ordinary or partial nonlinear differential equations which are algebraic in the indeterminates and their derivatives. We call these solutions D-algebraic functions, and their equations…
In one of his work, appeared in 1969, John A. Baker initiated the systematic investigation of some partial difference equations. The main purpose of this paper is to continue and to extend these investigations. Firstly, we present how such…
We consider the problem of solving partial differential equations (PDEs) in domains with complex microparticle geometry that is impractical, or intractable, to model explicitly. Drawing inspiration from volume rendering, we propose tackling…
The aim of this work is to derive new explicit solutions to the $\infty$-Laplace equation, the fundamental PDE arising in Calculus of Variations in the space $L^\infty$. These solutions obey certain symmetry conditions and are derived in…
Developing algorithms for solving high-dimensional partial differential equations (PDEs) has been an exceedingly difficult task for a long time, due to the notoriously difficult problem known as the "curse of dimensionality". This paper…
Parabolic partial differential equations (PDEs) appear in many disciplines to model the evolution of various mathematical objects, such as probability flows, value functions in control theory, and derivative prices in finance. It is often…
The notion of Laplace invariants is transferred to the lattices and discrete equations which are difference analogs of hyperbolic PDE's with two independent variables. The sequence of Laplace invariants satisfy the discrete analog of…
There are many numerical methods for solving partial different equations (PDEs) on manifolds such as classical implicit, finite difference, finite element, and isogeometric analysis methods which aim at improving the interoperability…
This article addresses linear hyperbolic partial differential equations with non-smooth coefficients and distributional data. Solutions are studied in the framework of Colombeau algebras of generalized functions. Its aim is to prove upper…
We consider a classical problem of Computer Algebra: symbolic solution of PDEs. We transform the famous Darboux theorems on differential transformations of hyperbolic operator into the space of invariants. We introduce a new idea -- $X$-…
We present the hyperasymptotic expansions for a certain group of solutions of the heat equation. We extend this result to a more general case of linear PDEs with constant coefficients. The generalisation is based on the method of Borel…
In this paper we present an algorithmic procedure that transforms, if possible, a given system of ordinary or partial differential equations with radical dependencies in the unknown function and its derivatives into a system with polynomial…
This is not a new result. Purpose of this work is to describe a method to search the analytical expression of the general real solution of the two-dimensional Laplace differential equation. This thing is not easy to find in scientific…
We describe a method of obtaining closed-form complete solutions of certain second-order linear partial differential equations with more than two independent variables. This method generalizes the classical method of Laplace transformations…