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

1966 ◽  
Vol 18 ◽  
pp. 1272-1280
Author(s):  
Josephine Mitchell
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Entire Functions ◽  
Integral Operators ◽  
Holomorphic Functions ◽  
Complex Variables ◽  
Partial Differential ◽  
Linear Partial Differential Equations ◽  
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.

XVI.—Simultaneous Linear Partial Differential Equations of the Second Order

1943 ◽  
Vol 61 (3) ◽  
pp. 195-209
Author(s):  
E. L. Ince
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Hypergeometric Function ◽  
Second Order ◽  
Partial Differential ◽  
Compatible System ◽  
Linear Partial Differential Equations ◽  
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.

On the Location of Singularities of a Class of Elliptic Partial Differential Equations in Four Variables

1965 ◽  
Vol 17 ◽  
pp. 676-686 ◽  
Author(s):  
R. P. Gilbert
Keyword(s):  
Partial Differential Equations ◽  
Elliptic Equation ◽  
Differential Equations ◽  
Complex Variable ◽  
Entire Functions ◽  
Elliptic Partial Differential Equations ◽  
Partial Differential ◽  
Singular Behaviour ◽  
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In this paper we shall investigate the singular behaviour of the solutions to the elliptic equation(1.1)where A (r2), C(r2) are entire functions of the complex variable

On Some Relations Between Partial and Ordinary Differential Equations

1958 ◽  
Vol 10 ◽  
pp. 183-190 ◽  
Author(s):  
Erwin Kreyszig
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Complex Variable ◽  
Analytic Functions ◽  
Integral Operators ◽  
Basic Tool ◽  
Partial Differential ◽  
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The theory of solutions of partial differential equations (1.1) with analytic coefficients can be based upon the theory of analytic functions of a complex variable; the basic tool in this approach is integral operators which map the set of solutions of (1.1) onto the algebra of analytic functions. For certain classes of operators this mapping which is first defined in the small, can be continued to the large, cf. Bergman (3).

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