THE APPLICATIONS IN ENGINEERING OF EQUATIONS WITH NON-STANDARD GROWTH CONDITIONAL

Year 2018, Volume: 13 Issue: 3, 167 - 179, 23.07.2018

Ebubekir Akkoyunlu , Rabil Ayazoğlu , Sezgin Akbulut

Abstract

Many materials and problems in physics and engineering applications can be
mathematically modeled with sufficient accuracy using classical Lebesgue  and classical Sobolev  spaces. However,  must be variable in order to be expressed correctly
the underlying energy of some nonhomogeneous materials. Such problems can be
solved only in the variable-exponent Lebesgue  and Sobolev  spaces. Therefore, in recent years, the
interest to partial differential equations with non-standard growth conditional
involving  -Laplacian (with  growth conditional) and variational integrals have
been increased. Electrorheological
Fluids Theory, Nonlinear Elasticity Theory, Image Processing, Flow in Porous
Media are  some of the application areas in engineering of non-standard growth conditional
differential equations involving -Laplacian.
Especially Electrorheological fluids have been used in
robotics and space technology (The Research is mostly done in America and
especially in NASA laboratories) have significiant importance. In this presentation, we provide information on variational integrals
and on nonstandard growth-conditional partial differential equations involving -Laplacian,
which has an important role in applied sciences (especially in engineering).

Keywords

: p(x)–Laplacian, Non-Standart Growth Condition, Variational Integral, Lebesgue, Sobolev

STANDART OLMAYAN BÜYÜME KOŞULLU DENKLEMLERİN MÜHENDİSLİKTEKİ UYGULAMALARI

Abstract

Fizik alanında ve
mühendislik uygulamalarında birçok materyal ve problem
klasik Lebesgue  ve klasik Sobolev uzayları kullanılarak
yeterli doğrulukla matematiksel olarak modellenebilir. Ancak bazı nonhomojen
materyallerin etkin enerjisinin doğru şekilde ifade edilebilmesi için üssünün
değişken olması gerekir. Bu tür problemlerin çözümleri yalnız değişken üslü Lebesgue  ve Sobolev  uzaylarında mümkündür. Bundan
dolayı son yıllarda, –Laplacian içeren standart olmayan büyüme koşullu ( büyüme koşullu) kısmi diferansiyel denklemlere
ve varyasyonel integrallere olan ilgi artmıştır. –Laplacian içeren standart olmayan büyüme koşullu diferansiyel
denklemlerin uygulama alanlarından bazıları electroreheolojik akışkanlar teorisi,
lineer olmayan esneklik teorisi, görüntü iyileştirme ve gözenekli ortamlarda akış’dır.
Bunlar içerisinde en önemlisi robot ve uzay
teknolojisinde de kullanılan 
(araştırmaları çoğunlukla Amerika’da ve özellikle NASA
laboratuvarlarında yapılan) electroreheolojik akışkanlar (ER akışkanlar)
teorisidir. Bu sunumumuzda uygulamalı bilimlerde (özellikle mühendislikte)
önemli bir yere sahip olan –Laplacian içeren standart olmayan büyüme koşullu kısmi
diferansiyel denklemler ve varyasyonel integrallerle ilgili çalışmalar hakkında
bilgi verilecektir.

Keywords

p(x)–Laplacian, Standart Olmayan Büyüme Koşulu, Varyasyonel İntegral, Lebesgue, Sobolev

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