Approximate Analysis to the KdV-Burgers Equation Zhaosheng Feng - PowerPoint PPT Presentation

Approximate Analysis to the KdV-Burgers Equation Zhaosheng Feng Department of Mathematics University of Texas-Pan American 1201 W. University Dr. Edinburg, Texas 78539, USA E-mail: zsfeng@utpa.edu October 26, 2013 Texas Analysis and Mathematical Physics Symposium—Rice University

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Outline Introduction 1 Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Outline Introduction 1 Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation Qualitative Analysis 2 Generalized Abel Equation Property of Our Operator Two Theorems KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Outline Introduction 1 Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation Qualitative Analysis 2 Generalized Abel Equation Property of Our Operator Two Theorems Approximate Solution 3 2D KdV-Burgers Equation Resultant Abel Equation Approximate Solution to 2D KdV-Burgers Equation KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Outline Introduction 1 Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation Qualitative Analysis 2 Generalized Abel Equation Property of Our Operator Two Theorems Approximate Solution 3 2D KdV-Burgers Equation Resultant Abel Equation Approximate Solution to 2D KdV-Burgers Equation Conclusion 4 KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Outline Introduction 1 Generalized KdV-Burgers Equation KdV-Burgers Equation Planar Polynomial Systems and Abel Equation Qualitative Analysis 2 Generalized Abel Equation Property of Our Operator Two Theorems Approximate Solution 3 2D KdV-Burgers Equation Resultant Abel Equation Approximate Solution to 2D KdV-Burgers Equation Conclusion 4 Acknowledgement 5 KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 2 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] � � δ u xx + β p u p u t + + α u x − µ u xx = 0 , (1) x where u is a function of x and t , α , β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] � � δ u xx + β p u p u t + + α u x − µ u xx = 0 , (1) x where u is a function of x and t , α , β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones. KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] � � δ u xx + β p u p u t + + α u x − µ u xx = 0 , (1) x where u is a function of x and t , α , β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones. The type of such models has the simplest form of wave equation in which nonlinearity ( u p ) x , dispersion u xxx and dissipation u xx all occur. KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] � � δ u xx + β p u p u t + + α u x − µ u xx = 0 , (1) x where u is a function of x and t , α , β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones. The type of such models has the simplest form of wave equation in which nonlinearity ( u p ) x , dispersion u xxx and dissipation u xx all occur. — — KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] � � δ u xx + β p u p u t + + α u x − µ u xx = 0 , (1) x where u is a function of x and t , α , β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones. The type of such models has the simplest form of wave equation in which nonlinearity ( u p ) x , dispersion u xxx and dissipation u xx all occur. — — [1] J.L. Bona, W.G. Pritchard and L.R. Scott, Philos. Trans. Roy. Soc. London Ser. A , 302 (1981), 457–510. KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Generalized KdV-Burgers Equation Generalized Korteweg-de Vries-Burgers equation [1, 2] � � δ u xx + β p u p u t + + α u x − µ u xx = 0 , (1) x where u is a function of x and t , α , β and p > 0 are real constants, µ and δ are coefficients of dissipation and dispersion, respectively. The type of such problems arises in modeling waves generated by a wavemaker in a channel and waves incoming from deep water into nearshore zones. The type of such models has the simplest form of wave equation in which nonlinearity ( u p ) x , dispersion u xxx and dissipation u xx all occur. — — [1] J.L. Bona, W.G. Pritchard and L.R. Scott, Philos. Trans. Roy. Soc. London Ser. A , 302 (1981), 457–510. [2] J.L.Bona, S.M. Sun and B.Y. Zhang, Dyn. Partial Differ. Equs . 3 (2006), 1–69. KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 3 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: u t + α uu x + β u xx = 0 , (2) KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: u t + α uu x + β u xx = 0 , (2) with the wave solution α + 2 β k u ( x , t ) = 2 k α tanh k ( x − 2 kt ) . KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: u t + α uu x + β u xx = 0 , (2) with the wave solution α + 2 β k u ( x , t ) = 2 k α tanh k ( x − 2 kt ) . Choices of α = µ = 0 and p = 2 lead (1) to the KdV equation [4]: u t + α uu x + su xxx = 0 , (3) KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26

Introduction Qualitative Analysis Approximate Solution Conclusion Acknowledgement Burgers Equation and KdV Equation Choices of δ = α = 0 and p = 2 lead (1) to the Burgers equation [3]: u t + α uu x + β u xx = 0 , (2) with the wave solution α + 2 β k u ( x , t ) = 2 k α tanh k ( x − 2 kt ) . Choices of α = µ = 0 and p = 2 lead (1) to the KdV equation [4]: u t + α uu x + su xxx = 0 , (3) with the soliton solution [5] u ( x , t ) = 12 sk 2 sech 2 k ( x − 4 sk 2 t ) . α KdV-Burgers Equation Z. Feng Department of Mathematics, University of Texas-Pan American, Edinburg, USA 4 / 26