A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid
We study the following initial-boundary value problem {ut − μt+αt(∂/∂t)∂2u/∂x2+(γ/x)(∂u/∂x) + fu = f1x,t, 1<x<R, t>0; u(1,t)=g1(t), u(R,t)=gR(t); u(x,0)=u~0(x)}, where γ>0,R>1 are given constants and f,f1,g1,gR,u~0,α, and μ are given functions. In Part 1, we use the Galerkin method an...
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| Format: | Article |
| Language: | English |
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Wiley
2016-01-01
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| Series: | Discrete Dynamics in Nature and Society |
| Online Access: | http://dx.doi.org/10.1155/2016/3875324 |
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| author | Le Thi Phuong Ngoc Truong Thi Nhan Nguyen Thanh Long |
| author_facet | Le Thi Phuong Ngoc Truong Thi Nhan Nguyen Thanh Long |
| author_sort | Le Thi Phuong Ngoc |
| collection | DOAJ |
| description | We study the following initial-boundary value problem {ut − μt+αt(∂/∂t)∂2u/∂x2+(γ/x)(∂u/∂x) + fu = f1x,t, 1<x<R, t>0; u(1,t)=g1(t), u(R,t)=gR(t); u(x,0)=u~0(x)}, where γ>0,R>1 are given constants and f,f1,g1,gR,u~0,α, and μ are given functions. In Part 1, we use the Galerkin method and compactness method to prove the existence of a unique weak solution of the problem above on (0,T), for every T>0. In Part 2, we investigate asymptotic behavior of the solution as t→+∞. In Part 3, we prove the existence and uniqueness of a weak solution of problem {ut − μt+αt(∂/∂t)∂2u/∂x2+(γ/x)(∂u/∂x) + fu = f1x,t, 1<x<R, t>0; u(1,t)=g1(t), u(R,t)=gR(t)} associated with a “(η,T)-periodic condition” u(x,0)=ηu(x,T), where 0<η≤1 is given constant. |
| format | Article |
| id | doaj-art-a2795226b33a4b68bab4b798bff75582 |
| institution | Kabale University |
| issn | 1026-0226 1607-887X |
| language | English |
| publishDate | 2016-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Discrete Dynamics in Nature and Society |
| spelling | doaj-art-a2795226b33a4b68bab4b798bff755822025-02-03T01:30:09ZengWileyDiscrete Dynamics in Nature and Society1026-02261607-887X2016-01-01201610.1155/2016/38753243875324A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade FluidLe Thi Phuong Ngoc0Truong Thi Nhan1Nguyen Thanh Long2University of Khanh Hoa, 01 Nguyen Chanh Str., Nha Trang, VietnamThe Faculty of Natural Basic Sciences, Vietnamese Naval Academy, 30 Tran Phu Street, Nha Trang, VietnamDepartment of Mathematics and Computer Science, University of Natural Science, Vietnam National University Ho Chi Minh City, 227 Nguyen Van Cu Str., Dist. 5, Ho Chi Minh City, VietnamWe study the following initial-boundary value problem {ut − μt+αt(∂/∂t)∂2u/∂x2+(γ/x)(∂u/∂x) + fu = f1x,t, 1<x<R, t>0; u(1,t)=g1(t), u(R,t)=gR(t); u(x,0)=u~0(x)}, where γ>0,R>1 are given constants and f,f1,g1,gR,u~0,α, and μ are given functions. In Part 1, we use the Galerkin method and compactness method to prove the existence of a unique weak solution of the problem above on (0,T), for every T>0. In Part 2, we investigate asymptotic behavior of the solution as t→+∞. In Part 3, we prove the existence and uniqueness of a weak solution of problem {ut − μt+αt(∂/∂t)∂2u/∂x2+(γ/x)(∂u/∂x) + fu = f1x,t, 1<x<R, t>0; u(1,t)=g1(t), u(R,t)=gR(t)} associated with a “(η,T)-periodic condition” u(x,0)=ηu(x,T), where 0<η≤1 is given constant.http://dx.doi.org/10.1155/2016/3875324 |
| spellingShingle | Le Thi Phuong Ngoc Truong Thi Nhan Nguyen Thanh Long A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid Discrete Dynamics in Nature and Society |
| title | A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid |
| title_full | A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid |
| title_fullStr | A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid |
| title_full_unstemmed | A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid |
| title_short | A Nonhomogeneous Dirichlet Problem for a Nonlinear Pseudoparabolic Equation Arising in the Flow of Second-Grade Fluid |
| title_sort | nonhomogeneous dirichlet problem for a nonlinear pseudoparabolic equation arising in the flow of second grade fluid |
| url | http://dx.doi.org/10.1155/2016/3875324 |
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