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In multi-phase fluid flow, fluid-structure interaction, and other applications, partial differential equations (PDEs) often arise with discontinuous coefficients and singular sources (e.g., Dirac delta functions). These complexities arise…
INTRODUCTION This papers deals with partial differential equations of second order, linear, with constant and not constant coefficients, in two variables, which admit real characteristics. I face the study of PDEs with the mentality of the…
We study the Lie point symmetries of a general class of partial differential equations (PDE) of second order. An equation from this class naturally defines a second-order symmetric tensor (metric). In the case the PDE is linear on the first…
Quantum computers can produce a quantum encoding of the solution of a system of differential equations exponentially faster than a classical algorithm can produce an explicit description. However, while high-precision quantum algorithms for…
For partial differential equations (PDEs) that have $n\geq2$ independent variables and a symmetry algebra of dimension at least $n-1$, an explicit algorithmic method is presented for finding all symmetry-invariant conservation laws that…
New solutions for second-order intertwining relations in two-dimensional SUSY QM are found via the repeated use of the first order supersymmetrical transformations with intermediate constant unitary rotation. Potentials obtained by this…
Partial differential equations (PDEs) are crucial for modeling various physical phenomena such as heat transfer, fluid flow, and electromagnetic waves. In computer-aided engineering (CAE), the ability to handle fine resolutions and large…
Two new approaches to solving first-order quasilinear elliptic systems of PDEs in many dimensions are proposed. The first method is based on an analysis of multimode solutions expressible in terms of Riemann invariants, based on links…
High-dimensional partial differential equations (PDE) appear in a number of models from the financial industry, such as in derivative pricing models, credit valuation adjustment (CVA) models, or portfolio optimization models. The PDEs in…
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 formulate a well-posedness and approximation theory for a class of generalised saddle point problems with a specific form of constraints. In this way we develop an approach to a class of fourth order elliptic partial differential…
Integration operational matrix methods based on Zernike polynomials are used to determine approximate solutions of a class of non-homogeneous partial differential equations (PDEs) of first and second order. Due to the nature of the Zernike…
An extension of the ideas of the Prelle-Singer procedure to second order differential equations is proposed. As in the original PS procedure, this version of our method deals with differential equations of the form…
Further investigations of implicit solutions to non-linear partial differential equations are pursued. Of particular interest are the equations which are Lorentz invariant. The question of which differential equations of second order for a…
We study the behaviour of solutions of ordinary differential equations of the second order with singular points, where the coefficients of the second-order derivative vanishes. In particular, we consider solutions entering a singular point…
A system of inhomogeneous second-order difference equations with linear parts given by noncommutative matrix coefficients are considered. Closed form of its solution is derived by means of newly defined delayed matrix sine/cosine using the…
We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…
Bagderina \cite{Bagderina2013} solved the equivalence problem for a family of scalar second-order ordinary differential equations (ODEs), with cubic nonlinearity in the first-order derivative, via point transformations. However, the…
Second-order partial differential equations in non-divergence form are considered. Equations of this kind typically arise as subproblems for the solution of Hamilton-Jacobi-Bellman equations in the context of stochastic optimal control, or…
An explicit formula to find symmetry recursion operators for partial differential equations (PDEs) is obtained from new results connecting variational integrating factors and non-variational symmetries. The formula is special case of a…