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

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

Öz

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.

Anahtar Kelimeler

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

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