Boundary Value Problems in Dimensions 7, 4 and 3 Related to Exceptional Holonomy

2018 ◽  
pp. 115-134 ◽  
Author(s):  
Simon Donaldson
Keyword(s):  
Variational Problem ◽  
Boundary Value Problems ◽  
General Framework ◽  
Boundary Value ◽  
Point Of View ◽  
The Real ◽  
Symmetry Reductions ◽  
Exceptional Holonomy ◽  
Monge Ampere

The variational point of view on exceptional structures in dimensions 6, 7 and 8 is one of Nigel Hitchin’s seminal contributions. One feature of this point of view is that it motivates the study of boundary value problems, for structures with prescribed data on a boundary. This chapter considers the case of 7 dimensions and G 2 structures. It briefly reviews a general framework and then goes on to examine in more detail symmetry reductions to dimensions 4 and 3. In the latter case, the chapter presents an interesting variational problem related to the real Monge–Ampère equation and describes a generalization of this.

scholarly journals On Solutions of Fractional Order Boundary Value Problems with Integral Boundary Conditions in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Hussein A. H. Salem ◽  
Mieczysław Cichoń
Keyword(s):  
Boundary Conditions ◽  
Boundary Value Problems ◽  
Fractional Order ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Integral Boundary Conditions ◽  
Point Of View ◽  
Integral Boundary ◽  
Special Cases

The object of this paper is to investigate the existence of a class of solutions for some boundary value problems of fractional order with integral boundary conditions. The considered problems are very interesting and important from an application point of view. They include two, three, multipoint, and nonlocal boundary value problems as special cases. We stress on single and multivalued problems for which the nonlinear term is assumed only to be Pettis integrable and depends on the fractional derivative of an unknown function. Some investigations on fractional Pettis integrability for functions and multifunctions are also presented. An example illustrating the main result is given.

scholarly journals Non-autonomous boundary value problems on the real line

2006 ◽  
Vol 15 (3) ◽  
pp. 759-776 ◽  
Author(s):  
Barbara Bianconi ◽  
◽  
Francesca Papalini
Keyword(s):  
Boundary Value Problems ◽  
Real Line ◽  
Boundary Value ◽  
The Real

scholarly journals APPROXIMATE SOLUTIONS OF SOME NONLINEAR PROBLEMS FOR THE MONGE-AMPERE EQUATIO

2020 ◽  
pp. 97-105
Author(s):  
N.B. Iskakova ◽  
◽  
А.S. Rysbek ◽  
N.S. Serik ◽  
◽  
...  
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Nonlinear Boundary ◽  
Initial Conditions ◽  
Boundary Value ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Mathcad Software ◽  
The Right ◽  
Monge Ampere

Due to numerous applications in various fields of science, including gas dynamics, meteorology, differential geometry, and others, the Monge – ampere equation is one of the most intensively studied equations of nonlinear mathematical physics.In this report, we study a nonlinear boundary value problem for the inhomogeneous Monge-ampere equation, the right part of which contains power nonlinearities in derivatives and arbitrary nonlinearity from the desired function.Based on linearization, the studied boundary value problems are reduced to a system of ordinary first-order differential equations with initial conditions that depend on the parameter.Methods for constructing exact and approximate solutions of some boundary value problems for the Monge-ampere equation are proposed.Using the Mathcad software package, numerical implementation of methods for constructing approximate solutions of the obtained systems of ordinary differential equations with a parameter is performed.Three-dimensional graphs of exact and approximate solutions of the problems under consideration in the Grafikus service are constructed.

scholarly journals On inversion in the case of the fundamentally finite integrable Vekua complex differential equation

2020 ◽  
Vol 108 (122) ◽  
pp. 13-22
Author(s):  
Milos Canak ◽  
Miloljub Albijanic
Keyword(s):  
Differential Equation ◽  
Fixed Point ◽  
General Solution ◽  
Boundary Value Problems ◽  
Imaginary Part ◽  
Boundary Value ◽  
The Real ◽  
Complex Differential Equation ◽  
The Core ◽  
Vekua Equation

The class of so called fundamentally finite integrable Vekua CDE is defined using the fixed point of the inversion and where one solution is equal to the coefficient of the equation. Then the different manifestations of inversion in relation to the general solution, an arbitrary analytical function inside and the core of the coefficient are examined. It shows that all the major problems of the Vekua equation theories, including boundary value problems can be interpreted and solved using the principle of inversion. The main significance of the fundamentally finite integrable Vekua equation is that the real and imaginary part of the solution can be separated, which in many mechanical and technique problems have certain physical meanings.

scholarly journals A critical point approach to boundary value problems on the real line

2018 ◽  
Vol 76 ◽  
pp. 215-220 ◽  
Author(s):  
Martin Bohner ◽  
Giuseppe Caristi ◽  
Shapour Heidarkhani ◽  
Shahin Moradi
Keyword(s):  
Boundary Value Problems ◽  
Critical Point ◽  
Real Line ◽  
Boundary Value ◽  
The Real
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