Chapter 1 Introduction 1
1.1 History and Motivation 1
1.2 Outline of the Work 4
Chapter 2 Hermite Pseudospectral Method and Its Application 6
2.1 Hermite Interpolation in One Dimension 6
2.2 Hermite Interpolation in Multiple Dimensions 14
2.3 Application to Burgers Equation on the Whole Line 23
2.4 Numerical Results 34
Chapter3 Laguerre Pseudospectral Method and Its Application 41
3.1 Laguerre-Gauss-Radau Interpolation in One Dimension 42
3.2 Laguerre Pseudospectral Method in One Dimension 53
3.3 Laguerre-Gauss-Radau Interpolation and Laguerre Pseudospectral Method in Multiple Dimensions 66
Chapter 4 On Two-Dimensional Incompressible Fluid Flow in an Infinite Tube 89
4.1 Introduction 89
4.2 Existence and Uniqueness of Global Weak Solution 91
4.3 Regularity of Solution 111
4.4 Other Problems 121
Chapter 5 Mixed Laguerre- Legendre Spectral Method for Incompressible Fluid Flow in an Infinite Tube 124
5.1 Mixed Laguerre- Legendre Approximation 124
5.2 Mixed Laguerre- Legendre Spectral Scheme 142
5.3 Numerical Results 150
Chapter 6 Mixed Laguerre-Legendre Pseudospectral Method for Incompressible Fluid Flow in an Infinite Tube 153
6.1 Mixed Laguerre-Legendre Interpolation Approximation 154
6.2 Mixed Laguerre-Legendre Pseudospectral Scheme 173
6.3 Proof of Error Estimates 180
Appendix A 202
Appendix B 209
Appendix C 212
References 215
Acknowledgement 221