A Cauchy integral formula for a class of elliptic systems of partial differential equations of first order in the space

1982 ◽  
Vol 108 (1) ◽  
pp. 167-178 ◽  
Author(s):  
Bernd Goldschmidt
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Integral Formula ◽  
Elliptic Systems ◽  
Cauchy Integral Formula ◽  
Cauchy Integral ◽  
Partial Differential ◽  
First Order

scholarly journals Quaternionic G-Monogenic Mappings in Em

2018 ◽  
Vol 12 ◽  
pp. 1-34
Author(s):  
Vitalii S. Shpakivskyi ◽  
Tetyana S. Kuzmenko
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Variable ◽  
Analytic Functions ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Cauchy Integral ◽  
Partial Differential ◽  
Morera Theorem ◽  
Classical Integral

We consider a class of so-called quaternionic G-monogenic mappings associatedwith m-dimensional (m 2 f2; 3; 4g) partial differential equations and propose a description of allmappings from this class by using four analytic functions of complex variable. For G-monogenicmappings we generalize some analogues of classical integral theorems of the holomorphic functiontheory of the complex variable (the surface and the curvilinear Cauchy integral theorems,the Cauchy integral formula, the Morera theorem), and Taylor’s and Laurent’s expansions.Moreover, we investigated the relation between G-monogenic and H-monogenic (differentiablein the sense of Hausdorff) quaternionic mappings.

scholarly journals A gradient technique for an optimal control problem governed by a system of nonlinear first order partial differential equations

1995 ◽  
Vol 36 (3) ◽  
pp. 261-273 ◽  
Author(s):  
Mohammad A. Kazemi
Keyword(s):  
Optimal Control ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Optimal Control Problems ◽  
Fréchet Derivative ◽  
Necessary Condition ◽  
Control Problems ◽  
Partial Differential ◽  
Frechet Derivative ◽  
First Order

AbstractIn this paper a class of optimal control problems with distributed parameters is considered. The governing equations are nonlinear first order partial differential equations that arise in the study of heterogeneous reactors and control of chemical processes. The main focus of the present paper is the mathematical theory underlying the algorithm. A conditional gradient method is used to devise an algorithm for solving such optimal control problems. A formula for the Fréchet derivative of the objective function is obtained, and its properties are studied. A necessary condition for optimality in terms of the Fréchet derivative is presented, and then it is shown that any accumulation point of the sequence of admissible controls generated by the algorithm satisfies this necessary condition for optimality.

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