Multidimensional Complex Analysis and Partial Differential Equations

1997 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Analysis ◽  
Partial Differential ◽  
Multidimensional Complex Analysis ◽  
Multidimensional Complex

scholarly journals On the theory and practice of thin-walled structures

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tamaz Vashakmadze
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Complex Analysis ◽  
Theory And Practice ◽  
Linear Case ◽  
Wide Sense ◽  
Partial Differential ◽  
Thin Walled Structures ◽  
Thin Walled

Abstract The basic problem of satisfaction of boundary conditions is considered when the generalized stress vector is given on the surfaces of elastic plates and shells. This problem has so far remained open for both refined theories in a wide sense and hierarchic type models. In the linear case, it was formulated by I. N. Vekua for hierarchic models. In the nonlinear case, bending and compression-expansion processes do not split and in this context the exact structure is presented for the system of differential equations of von Kármán–Mindlin–Reisner (KMR) type, constructed without using a variety of ad hoc assumptions since one of the two relations of this system in the classical form is the compatibility condition, but not the equilibrium equation. In this paper, a unity mathematical theory is elaborated in both linear and nonlinear cases for anisotropic inhomogeneous elastic thin-walled structures. The theory approximately satisfies the corresponding system of partial differential equations and the boundary conditions on the surfaces of such structures. The problem is investigated and solved for hierarchic models too. The obtained results broaden the sphere of applications of complex analysis methods. The classical theory of finding a general solution of partial differential equations of complex analysis, which in the linear case was thoroughly developed in the works of Goursat, Weyl, Walsh, Bergman, Kolosov, Muskhelishvili, Bers, Vekua and others, is extended to the solution of basic nonlinear differential equations containing the nonlinear summand, which is a composition of Laplace and Monge–Ampére operators.

Partial Differential Equations and Complex Analysis

2018 ◽  
Author(s):  
Steven G. Krantz ◽  
Estela A. Gavosto ◽  
Marco M. Peloso
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Analysis ◽  
Partial Differential

A hybrid analytical–numerical method for solving evolution partial differential equations. I. The half-line

2008 ◽  
Vol 464 (2095) ◽  
pp. 1823-1849 ◽  
Author(s):  
N Flyer ◽  
A.S Fokas
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Plane ◽  
Complex Analysis ◽  
Variable Coefficients ◽  
Integral Representations ◽  
New Method ◽  
Half Line ◽  
Time Step ◽  
Partial Differential

A new method, combining complex analysis with numerics, is introduced for solving a large class of linear partial differential equations (PDEs). This includes any linear constant coefficient PDE, as well as a limited class of PDEs with variable coefficients (such as the Laplace and the Helmholtz equations in cylindrical coordinates). The method yields novel integral representations, even for the solution of classical problems that would normally be solved via the Fourier or Laplace transforms. Examples include the heat equation and the first and second versions of the Stokes equation for arbitrary initial and boundary data on the half-line. The new method has advantages in comparison with classical methods, such as avoiding the solution of ordinary differential equations that result from the classical transforms, as well as constructing integral solutions in the complex plane which converge exponentially fast and which are uniformly convergent at the boundaries. As a result, these solutions are well suited for numerics, allowing the solution to be computed at any point in space and time without the need to time step. Simple deformation of the contours of integration followed by mapping the contours from the complex plane to the real line allow for fast and efficient numerical evaluation of the integrals.

scholarly journals On the rigorous foundations of the Fokas method for linear elliptic partial differential equations

2012 ◽  
Vol 468 (2141) ◽  
pp. 1325-1331 ◽  
Author(s):  
A. C. L. Ashton
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Complex Analysis ◽  
Boundary Value ◽  
Elliptic Partial Differential Equations ◽  
Unknown Boundary ◽  
Partial Differential ◽  
Global Relation ◽  
Fokas Method

In this paper, we address some of the rigorous foundations of the Fokas method, confining attention to boundary value problems for linear elliptic partial differential equations on bounded convex domains. The central object in the method is the global relation, which is an integral equation in the spectral Fourier space that couples the given boundary data with the unknown boundary values. Using techniques from complex analysis of several variables, we prove that a solution to the global relation provides a solution to the corresponding boundary value problem, and that the solution to the global relation is unique. The result holds for any number of spatial dimensions and for a variety of boundary value problems.

scholarly journals Jean Leray. 7 November 1906 — 10 November 1998

2006 ◽  
Vol 52 ◽  
pp. 137-148
Author(s):  
Martin Andler
Keyword(s):  
World War Ii ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Complex Analysis ◽  
Applied Mathematics ◽  
Spectral Sequences ◽  
Sheaf Theory ◽  
Partial Differential ◽  
Navier Stokes Equation ◽  
Pure Mathematics

Jean Leray was one of the major mathematicians of the twentieth century. His primary focus in mathematics came from applications; indeed, a majority of his contributions were in the theory of partial differential equations arising in physics, notably his 1934 paper on the Navier–Stokes equation. World War II, during which he was a prisoner–of–war in Austria for five years, induced him to turn to pure mathematics to avoid helping the German war effort. There he worked in topology, developing two radically new ideas: sheaf theory and spectral sequences. After 1950 he came back to partial differential equations and became interested in complex analysis, writing a remarkable series of papers on the Cauchy problem. Leray remained mathematically active until the end of his life; in the course of his career he wrote 132 papers. His influence on present mathematics is tremendous. On the one hand, sheaf theory and spectral sequences became essential tools in contemporary pure mathematics, reaching far beyond their initial scope in topology. On the other hand, Leray can rightly be considered the intellectual guide of the distinguished French school of applied mathematics.

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