相关论文: Implicit Solutions of PDE's

A theorem providing necessary conditions enabling one to map a nonlinear system of first order partial differential equations to an equivalent first order autonomous and homogeneous quasilinear system is given. The reduction to quasilinear…

We provide the structure of regular/singular fast/slow decay radially symmetric solutions for a class of superlinear elliptic equations with an in- definite weight on the nonlinearity f (u, r). In particular we are interested in the case…

The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately…

A theorem that constructs a path integral solution for general second order partial differential equations is specialized to obtain path integrals that are solutions of elliptic, parabolic, and hyperbolic linear second order partial…

We find implicit general solutions for modified eikonal equations $u_a u_a=F(u_t)$, where lower indices at dependent variables designate derivatives, $a=1,2,..,n$, and summation is implied over the repeated indices. We will consider the…

In this paper we develop a systematic reduction procedure for determining intermediate integrals of second order hyperbolic equations so that exact solutions of the second order PDEs under interest can be obtained by solving first order…

In this paper we show that an arbitrary solution of one ordinary difference equation is also a solution for a hierarchy of integrable difference equations. We also provide an example of such a solution that is related to sequence generated…

This article deals with the asymptotic behavior of fourth order differential equation where the coefficients are perturbations of linear constant coefficient equation. We introduce a change of variable and deduce that the new variable…

We consider time-inhomogeneous, second order linear parabolic partial differential equations of the non-divergence type, and assume the ellipticity and the continuity on the coefficient of the second order derivatives and the boundedness on…

The nonlinear eigenvalue problem of a class of second order semi-transcendental differential equations is studied. A nonlinear eigenvalue is defined as the initial condition which gives rise a separatrix solution. A semi-transcendental…

We consider an equation of multiple variables in which a partial derivative does not vanish at a point. The implicit function theorem provides a local existence and uniqueness of the function for the equation. In this paper, we propose an…

Asymptotic formula is derived for the behavior of the fundamental solution of the second-order elliptic self-adjoint operator with a piecewise-smooth coefficient in front of the senior derivatives near the discontinuity surface of the…

In this paper, we study the Dirichlet problem for the implicit degen- erate nonlinear elliptic equation with variable exponent in a bounded domain. We obtain sufficient conditions for the existence of a solution with- out regularization and…

In this paper we consider second order perturbations of a flat Friedmann-Lema\^{i}tre universe whose stress-energy content is a single minimally coupled scalar field with an arbitrary potential. We derive the general solution of the…

The numerical solution of differential equations can be formulated as an inference problem to which formal statistical approaches can be applied. However, nonlinear partial differential equations (PDEs) pose substantial challenges from an…

Selection of 25 examples from extensive nontrivial families for different types of nonlinear PDEs and their formal general solutions are given. The main goal here is to show on examples the types of solvable PDEs and what their general…

Considering differential equation f''+A(z)f'+B(z)f=0, where A(z) and B(z) are entire complex functions, our results revolve around proving all non-trivial solutions are of infinite order taking various restrictions on coefficients A(z) and…

We analyze the polynomial solutions of a nonlinear integral equation, generalizing the work of C. Bender and E. Ben-Naim. We show that, in some cases, an orthogonal solution exists and we give its general form in terms of kernel…

We obtain several higher order exact periodic solutions of (i) a coupled symmetric phi4 model in an external field, (ii) an asymmetric coupled phi4 model, (iii) an asymmetric-symmetric coupled phi4 model, in terms of Lame polynomials of…

It is shown that globally positive solutions of a linear second order parabolic partial differential equation on a bounded domain, with Robin boundary conditions, are unique up to multiplication by a positive constant.