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

Yıl 2018, Cilt: 13 Sayı: 3, 167 - 179, 23.07.2018

Ebubekir Akkoyunlu , Rabil Ayazoğlu , Sezgin Akbulut

Öz

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).

Anahtar Kelimeler

: 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

Kaynakça

• Aboulaich, R., Meskine, D. and Souissi, A., (2008). New Diffusion Models in Image Processing. Computers&Mathematics with Applications, Volume:56, Number:4, pp:874–882.
• Acerbi, E. and Mingione, G., (2002). Regularity Results for Stationary Electrorheological Rluids. Archive for Rational Mechanics and Analysis, Volume:164, Number:3, pp:213-259.
• Antontsev, S.N. and Shmarev, S.I., (2005). On Localization of Solutions of Elliptic Equations with Nonhomogeneous Anisotropic Degeneracy. Siberian Mathematical Journal, Volume:46, Number:5, pp:765-782.
• Atkin, R.J., Shi, X. and Bullough, W.A., (1991). Solutions of the Constitutive Equations for the Flow of an Electrorheological Fluid in Radial Configurations. Journal of Rheology, Volume:35, Number:7, pp:1441-1461.
• Ayazoglu, R. and Ekincioglu, I., (2016). Electrorheological Fluids Equations Involving Variable Exponent with Dependence on the Gradient via Mountain Pass Techniques. Numerical Functional Analysis and Optimization, Volume:37, Number:9, pp: 1144-1157.
• Bailey, P., Gillies, D.G., Heyes, D.M. and Sutcliffe, L.H., (1989). Experimental and Simulation Studies of Electrorheology. Molecular Simulation, Volume:4, Number:1-3, pp: 137-151.
• Blomgren, P., Chan, T. F., Mulet, P. and Wong, C. K., (1997). Total Variation Image Restoration: Numerical Methods and Extensions. In Proceedings of the IEEE International Conference on Image Processing, Vol. III, IEEE, Los Alamitos, CA, pp: 384–387.
• Bollt, E.M., Chartrand, R., Esedoglu, S., Schultz P. and Vixie, K.R., (2009). Graduated Adaptive Image Denoising: Local Compromise between Total Variation and Isotropic Diffusion. Advances in Computational Mathematics, Volume:31, Number:1, pp: 61-85.
• Bonnecaze, R.T. and Brady, J.F., (1992). Yield Stresses in Electrorheological Fluids. Journal of Rheology, Volume:36, Number:1, pp: 73-115.
• Chambolle, A. and Lions, P. L., (1997). Image Recovery via Total Variation Minimization and Related Problems. Numerische Mathematik, Volume:76, Number:2, pp: 167–188.
• Chen, Y., Levine, S., Rao, M., (2006). Variable Exponent, Linear Growth Functionals in Image Restoration. SIAM Journal on Applied Mathematics, Volume:66, Number:4, pp: 1383–1406.
• Diening, L., Harjuletho, P., Hästö, P. and Růžička, M., (2011). Lebesgue and Sobolev Spaces with Variable Exponents. Springer.
• Grady, N., (2009). Functions of Bounded Variation. Dostopno. Prek:https://www.whitman.edu/Documents/Academics/Mathematics/grady. pdf (Dostopano: 7.2.2017).
• Halsey, T.C. and Will, T., (1990). Structure of Electrorheological Fluids. Physical Review Letters, Volume:65, Number:22, pp: 2820.
• Halsey, T.C., (1992). Electrorheological Fluids. Science, Volume:258, Number:5083, pp: 761-766.
• Harjulehto, P., Hästö, P., Latvala, V. and Toivanen, O., (2013). Critical Variable Exponent Functionals in Image Restoration. Applied Mathematics Letters, Volume:26, Number:1, pp: 56-60.
• Klingenberg, D.J., Frank, V.S. and Zukoski, C.F., (1991). The Small Shear Rate Response of Electrorheological Suspensions. II. Extension Beyond the Point–Dipole Limit. The Journal of Chemical Physics, Volume:94, Number:9, pp: 6170-6178.
• Kovăčik, O., Răkosnik, J., (1991). On the Space L^(p(x)) (Ω) and W^(k,p(x)) (Ω). Czechoslovak Math. J., Volume:41, Number:4, pp: 592-618.
• Levine, S., (2005). An Adaptive Variational Model for Image Decomposition, in: Energy Minimization Methods in Computer Vision and Pattern Recognition. Springer-Verlag, LCNS No. 3757, pp: 382–397.
• Li, F., Li, Z. and Pi, L., (2010). Variable Exponent Functionals in Image Restoration. Applied Mathematics and Computation, Volume:216, Number:3, pp: 870–882.
• Liu, B. and Li, F., (2012). Non-Simultaneous Blowup in Heat Equations with Nonstandard Growth Conditions. Journal of Differential Equations, Volume:252, Number:8, pp: 4481-4502.
• Mihâilescu, M., (2006). Elliptic Problems in Variable Exponent Spaces. Bulletin Australian Mathematical Society, Volume:74, Number:2, pp: 197-206.
• Mihăilescu, M., Rădulescu, V., (2006). A Multiplicity Result for a Nonlinear Degenerate Problem Arising in the Theory of Electrorheological Fluids. In Proceed¬ings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Volume:462, Number:2073, pp: 2625-2641.
• Nikolova, M., (2004). Weakly Constrained Minimization: Application to the Estimation of Images and Signals Involving Constant Regions. Journal of Mathematical Imaging and Vision, Volume: 21, Number:2, pp: 155–175.
• Orlicz, W., (1931). Uber Konjugierte Exponentenfolgen. Studia Math., Volume: 3, Number: 1, pp: 200-212.
• Parthasarathy, M. and Klingenberg, D.J., (1996). Electrorheology: Mechanisms and Models. Materials Science and Engineering: R: Reports, Volume:17, Number:2, pp: 57-103.
• Rajagopal, K.R. and Alan Wineman, S., (1992). Flow of Electro-rheological Materials. Acta Mechanica, Volume:91, Number:1-2, pp: 57-75.
• Rajagopal, K.R., Yalamanchili, R.C. and Wineman, A.S., (1994). Modeling Electrorheological Materials through Mixture Theory. International Journal of Engineering Science, Volume:32, Number:3, pp: 481-500.
• Rajagopal, K.R. and Růzǐčka, M., (1996). On the Modeling of Electrorheological Materials. Mechanics Research Communications, Volume:23, Number:4, pp: 401-407.
• Rudin, L.I., Osher, S. and Fatemi, E., (1992). Nonlinear Total Variation Based Noise Removal Algorithms. Nonlinear Phenomena, Volume:60, Number:1-4, pp: 259-268.
• Růžička, M., (2000). Electrorheological Fluids: Modeling and Mathemat¬ical Theory. Lecture Notes in Mathematics. Berlin: Springer-Verlag.
• Sharapudinov, I. I., (1979). Topology of the Space L^(p(t)) [0,1]. Matematicheskie Zametki, Volume:26, Number:4, pp: 613-632.
• Strong, D.M. and Chan, T.F., (1996). Spatially and Scale Adaptive Total Variation Based Regularization and Anisotropic Diffusion in Image Processing. Technical Report CAM96-46, University of California, Los Angeles, CA, Available online at http://www.math. ucla.edu/applied/cam/index.html.
• Tsenov, I.V., (1961). Generalization of the Problem of Best Approximation of a Function in the Space L^s. Uch. Zap. Dagestan Gos. Univ., Volume: 7, pp: 25¬-37.
• Whittle, M., (1990). Computer Simulation of an Electrorheological Fluid. Journal of Non-Newtonian Fluid Mechanics, Volume:37, Number:2-3, pp: 233-263.
• Wineman, A. S. and Rajagopal, K.R., (1995). On Constitutive Equations for Electrorheological Materials. Continuum Mechanics and Thermodynamics, Volume:7, Number:1, pp: 1-22.
• Winslow, W. M., (1949). Induced Fibration of Suspensions. Journal of Applied Physics, Volume:20, Number:12, pp: 1137-1140.
• Zhikov, V. V. E., (1987). Averaging of Functionals of the Calculus of Variations and Elasticity Theory. Izvestiya: Mathematics, Volume: 29, Number: 1, pp: 33-36.