5. Note on Linear Partial Differential Equations

Differential Equations ◽

Hypergeometric Function ◽

Second Order ◽

Partial Differential ◽

Compatible System ◽

Polynomial Coefficients ◽

Linearly Independent ◽

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SummaryIn the system of two linear partial differential equations of the second ordera,…,f were supposed to be polynomials in x, and a1…, f1 polynomials in y. These polynomial coefficients were subjected to certain restrictions, including conditions for the system having exactly four linearly independent solutions, and conditions for preserving the symmetrical aspect, in x and y, of the system. It has been proved that any compatible system of the contemplated form whose coefficients satisfy the stipulated conditions is equivalent with, i.e. transformable into, a hypergeometric system. More particularly it has been shown that the hypergeometric systems involved are the system of partial differential equations associated with Appell's hypergeometric function in two variables F2 and the confluent systems arising herefrom.

The Cauchy Problem For Linear Partial Differential Equations With Restricted Boundary Conditions

Canadian Journal of Mathematics ◽

10.4153/cjm-1956-050-5 ◽

1956 ◽

Vol 8 ◽

pp. 426-431 ◽

Cited By ~ 2

Author(s):

E. P. Miles ◽

Ernest Williams

Keyword(s):

Cauchy Problem ◽

Differential Operator ◽

Differential Equations ◽

Boundary Conditions ◽

Cauchy Data ◽

Partial Differential ◽

The Cauchy Problem ◽

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We shall discuss solutions of linear partial differential equations of the form1where Ψ is an ordinary differential operator of order s with respect to t. Our first theorem gives a solution of (1) for the Cauchy data;2j = 1,2, ߪ,s − 1,whenever the function P is annihilated by a finite iteration of the operator Φ.

Bounds for Solutions of a System of Linear Partial Differential Equations on Domains with Bergman-Silov Boundary

Canadian Journal of Mathematics ◽

10.4153/cjm-1966-126-8 ◽

1966 ◽

Vol 18 ◽

pp. 1272-1280

Author(s):

Josephine Mitchell

Keyword(s):

Differential Equations ◽

Entire Functions ◽

Integral Operators ◽

Holomorphic Functions ◽

Complex Variables ◽

Partial Differential ◽

Systems Of Differential Equations ◽

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The method of integral operators has been used by Bergman and others (4; 6; 7; 10; 12) to obtain many properties of solutions of linear partial differential equations. In the case of equations in two variables with entire coefficients various integral operators have been introduced which transform holomorphic functions of one complex variable into solutions of the equation. This approach has been extended to differential equations in more variables and systems of differential equations. Recently Bergman (6; 4) obtained an integral operator transforming certain combinations of holomorphic functions of two complex variables into functions of four real variables which are the real parts of solutions of the system1where z1, z1*, z2, z2* are independent complex variables and the functions Fj (J = 1, 2) are entire functions of the indicated variables.

XXII.—An Operational Method for the Solution of Linear Partial Differential Equations

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600023129 ◽

1932 ◽

Vol 51 ◽

pp. 176-189 ◽

Cited By ~ 1

Author(s):

W. O. Kermack ◽

W. H. McCrea

Keyword(s):

Differential Equations ◽

Contact Transformation ◽

Operational Method ◽

Partial Differential ◽

Definite Integrals ◽

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General Method

1. A general method for the solution of differential equations by definite integrals has recently been given by Professor E. T. Whittaker. It is briefly that, if a contact transformation from variables (q, p) to (Q, P) be given by Q = Q(q, p), P = P(q, p), and if this transforms an expression G(Q, P) into F(q, p), then the solutions of the differential equationsare connected by a relation of the form

On continuation of Gevrey class solutions of linear partial differential equations

Proceedings of the seventh International Colloquium on Differential Equations ◽

10.1515/9783112319185-027 ◽

1997 ◽

pp. 197-204

Keyword(s):

Linear Partial Differential Equations

Differential Equations ◽

Gevrey Class ◽

Partial Differential ◽

Lyapunov-type inequalities for partial differential equations with 𝑝-Laplacian

Forum Mathematicum ◽

10.1515/forum-2020-0232 ◽

2021 ◽

Vol 0 (0) ◽

Author(s):

Robert Stegliński

Keyword(s):

Differential Equations ◽

Boundary Conditions ◽

Dirichlet Boundary ◽

Nonlinear Partial Differential Equations ◽

Dirichlet Boundary Conditions ◽

Partial Differential ◽

Lyapunov Inequalities ◽

Funct Anal

Abstract The aim of this paper is to extend results from [A. Cañada, J. A. Montero and S. Villegas, Lyapunov inequalities for partial differential equations, J. Funct. Anal. 237 (2006), 1, 176–193] about Lyapunov-type inequalities for linear partial differential equations to nonlinear partial differential equations with 𝑝-Laplacian with zero Neumann or Dirichlet boundary conditions.