Partial Differential Equations and Applications: A Bridge for Students and Researchers in Applied Sciences offers a unique approach to this key subject by connecting mathematical principles to the latest research advances in select topics. Beginning with very elementary PDEs, such as classical heat equations, wave equations and Laplace equations, the book focuses on concrete examples. It gives students basic skills and techniques to find explicit solutions for partial differential equations.
As it progresses, the book covers more advanced topics such as the maximum principle and applications, Green’s representation, Schauder’s theory, finite-time blowup, and shock waves. By exploring these topics, students gain the necessary tools to deal with research topics in their own fields, whether proceeding in math or engineering areas.
1. 1.1: Introduction - p. 1 2. 1.2: Fundamental PDE questions: Well-posedness and ill-posedness - p. 3 3. 1.3: Some important equations and systems - p. 8 4. 1.4: Classification of second-order partial differential equations - p. 15 5. 1.5: Some elementary formulas and inequalities - p. 19 6. 1.6: Notes and remarks - p. 24 7. 1.7: Exercises - p. 25 8. References - p. 27 9. Chapter 2 Function spaces and the Fredholm Alternative - p. 29 10. 2.1: Banach spaces and Hilbert spaces - p. 29 11. 2.2: Function spaces - p. 31 12. 2.3: Completeness and series representation - p. 40 13. 2.4: Fourier series - p. 44 14. 2.5: The contraction mapping principle and applications - p. 48 15. 2.6: The continuity method - p. 52 16. 2.7: The Fredholm Alternative and applications - p. 54 17. 2.8: The Riesz representation and the Lax–Milgram theorem - p. 57 18. 2.9: Notes and remarks - p. 59 19. 2.10: Exercises - p. 59 20. References - p. 61 21. Chapter 3 Eigenvalue problems and eigenfunction expansions - p. 63 22. 3.1: The method of separation of variables - p. 63 23. 3.2: The Sturm–Liouville theory - p. 64 24. 3.3: The main theorem in the Sturm–Liouville theory - p. 70 25. 3.4: Eigenvalues problems in several space dimensions - p. 76 26. 3.5: Boundary eigenvalue problems - p. 80 27. 3.6: Convergence of function series - p. 83 28. 3.7: Notes and remarks - p. 84 29. 3.8: Exercises - p. 85 30. References - p. 87 31. Chapter 4 The heat equation - p. 89 32. 4.1: The mathematical model of heat conduction - p. 89 33. 4.2: Solution of the heat equation in one-space dimension - p. 92 34. 4.3: Solution of the heat equation in higher-space dimension - p. 99 35. 4.4: The well-posedness and energy estimates - p. 102 36. 4.5: A qualitative property: The maximum principle - p. 107 37. 4.6: Long-time behaviors of solutions - p. 111 38. 4.7: The comparison principle - p. 117 39. 4.8: Notes and remarks - p. 120 40. 4.9: Exercises - p. 120 41. References - p. 124 42. Chapter 5 The wave equation - p. 125 43. 5.1: The mathematical model of a vibrating string - p. 125 44. 5.2: Solutions of the wave equation in one-space dimension - p. 129 45. 5.3: Wave propagation in several space dimensions - p. 137 46. 5.4: Solution of the wave equation in higher space dimension - p. 142 47. 5.5: The well-posedness and energy estimates - p. 146 48. 5.6: A qualitative property: The finite propagation speed - p. 150 49. 5.7: The Cauchy problem for the wave equation - p. 151 50. 5.8: Notes and remarks - p. 161 51. 5.9: Exercises - p. 161 52. References - p. 164 53. Chapter 6 The Laplace equation - p. 165 54. 6.1: Some mathematical models of the Laplace equation - p. 165 55. 6.2: Series solution for the Laplace equation - p. 168 56. 6.3: The well-posedness of the Laplace equation - p. 180 57. 6.4: A qualitative property: The mean-value formula - p. 183 58. 6.5: The maximum principle and applications - p. 185 59. 6.6: An L∞(Ω)-estimate: Moser's iteration method - p. 189 60. 6.7: Notes and remarks - p. 192 61. 6.8: Exercises - p. 192 62. References - p. 195 63. Chapter 7 The Fourier transform and applications - p. 197 64. 7.1: Definition of the Fourier transform - p. 197 65. 7.2: Properties of the Fourier transform - p. 202 66. 7.3: Applications to the Laplace equation - p. 205 67. 7.4: Applications to the heat equation - p. 208 68. 7.5: Application to the wave equation - p. 212 69. 7.6: Generalized Fourier transform - p. 217 70. 7.7: Notes and remarks - p. 222 71. 7.8: Exercises - p. 222 72. References - p. 223 73. Chapter 8 The fundamental solution and Green's representation - p. 225 74. 8.1: Introduction to the fundamental solution - p. 225 75. 8.2: The fundamental solution of the Laplace equation - p. 228 76. 8.3: Green's functions for the Laplace operator - p. 233 77. 8.4: The fundamental solution of the heat equation - p. 243 78. 8.5: Green's function of the heat equation - p. 252 79. 8.6: Finite-time blowup and extinction - p. 255 80. 8.7: Notes and remarks - p. 265 81. 8.8: Exercises - p. 265 82. References - p. 267 83. Chapter 9 Systems of first-order partial differential equations - p. 269 84. 9.1: Some first-order PDE models in physical sciences - p. 269 85. 9.2: The characteristic method - p. 272 86. 9.3: Well-posedness for a system of first-order PDEs - p. 276 87. 9.4: The solution of Maxwell's equations - p. 278 88. 9.5: The conservation law and shock waves - p. 282 89. 9.6: A class of semilinear wave equations - p. 289 90. 9.7: Remarks and notes - p. 296 91. 9.8: Exercises - p. 296 92. References - p. 298 93. Appendix A Some essential results in ordinary differential equations - p. 299 94. Appendix B Sobolev spaces - p. 305 95. References - p. 309 96. Index - p. 311
Partial Differential Equations and Applications: A Bridge for Students and Researchers in Applied Sciences offers a unique approach to this key subject by connecting mathematical principles to the latest research advances in select topics. Beginning with very elementary PDEs, such as classical heat equations, wave equations and Laplace equations, the book focuses on concrete examples. It gives students basic skills and techniques to find explicit solutions for partial differential equations.
As it progresses, the book covers more advanced topics such as the maximum principle and applications, Green’s representation, Schauder’s theory, finite-time blowup, and shock waves. By exploring these topics, students gain the necessary tools to deal with research topics in their own fields, whether proceeding in math or engineering areas.
1. 1.1: Introduction - p. 1 2. 1.2: Fundamental PDE questions: Well-posedness and ill-posedness - p. 3 3. 1.3: Some important equations and systems - p. 8 4. 1.4: Classification of second-order partial differential equations - p. 15 5. 1.5: Some elementary formulas and inequalities - p. 19 6. 1.6: Notes and remarks - p. 24 7. 1.7: Exercises - p. 25 8. References - p. 27 9. Chapter 2 Function spaces and the Fredholm Alternative - p. 29 10. 2.1: Banach spaces and Hilbert spaces - p. 29 11. 2.2: Function spaces - p. 31 12. 2.3: Completeness and series representation - p. 40 13. 2.4: Fourier series - p. 44 14. 2.5: The contraction mapping principle and applications - p. 48 15. 2.6: The continuity method - p. 52 16. 2.7: The Fredholm Alternative and applications - p. 54 17. 2.8: The Riesz representation and the Lax–Milgram theorem - p. 57 18. 2.9: Notes and remarks - p. 59 19. 2.10: Exercises - p. 59 20. References - p. 61 21. Chapter 3 Eigenvalue problems and eigenfunction expansions - p. 63 22. 3.1: The method of separation of variables - p. 63 23. 3.2: The Sturm–Liouville theory - p. 64 24. 3.3: The main theorem in the Sturm–Liouville theory - p. 70 25. 3.4: Eigenvalues problems in several space dimensions - p. 76 26. 3.5: Boundary eigenvalue problems - p. 80 27. 3.6: Convergence of function series - p. 83 28. 3.7: Notes and remarks - p. 84 29. 3.8: Exercises - p. 85 30. References - p. 87 31. Chapter 4 The heat equation - p. 89 32. 4.1: The mathematical model of heat conduction - p. 89 33. 4.2: Solution of the heat equation in one-space dimension - p. 92 34. 4.3: Solution of the heat equation in higher-space dimension - p. 99 35. 4.4: The well-posedness and energy estimates - p. 102 36. 4.5: A qualitative property: The maximum principle - p. 107 37. 4.6: Long-time behaviors of solutions - p. 111 38. 4.7: The comparison principle - p. 117 39. 4.8: Notes and remarks - p. 120 40. 4.9: Exercises - p. 120 41. References - p. 124 42. Chapter 5 The wave equation - p. 125 43. 5.1: The mathematical model of a vibrating string - p. 125 44. 5.2: Solutions of the wave equation in one-space dimension - p. 129 45. 5.3: Wave propagation in several space dimensions - p. 137 46. 5.4: Solution of the wave equation in higher space dimension - p. 142 47. 5.5: The well-posedness and energy estimates - p. 146 48. 5.6: A qualitative property: The finite propagation speed - p. 150 49. 5.7: The Cauchy problem for the wave equation - p. 151 50. 5.8: Notes and remarks - p. 161 51. 5.9: Exercises - p. 161 52. References - p. 164 53. Chapter 6 The Laplace equation - p. 165 54. 6.1: Some mathematical models of the Laplace equation - p. 165 55. 6.2: Series solution for the Laplace equation - p. 168 56. 6.3: The well-posedness of the Laplace equation - p. 180 57. 6.4: A qualitative property: The mean-value formula - p. 183 58. 6.5: The maximum principle and applications - p. 185 59. 6.6: An L∞(Ω)-estimate: Moser's iteration method - p. 189 60. 6.7: Notes and remarks - p. 192 61. 6.8: Exercises - p. 192 62. References - p. 195 63. Chapter 7 The Fourier transform and applications - p. 197 64. 7.1: Definition of the Fourier transform - p. 197 65. 7.2: Properties of the Fourier transform - p. 202 66. 7.3: Applications to the Laplace equation - p. 205 67. 7.4: Applications to the heat equation - p. 208 68. 7.5: Application to the wave equation - p. 212 69. 7.6: Generalized Fourier transform - p. 217 70. 7.7: Notes and remarks - p. 222 71. 7.8: Exercises - p. 222 72. References - p. 223 73. Chapter 8 The fundamental solution and Green's representation - p. 225 74. 8.1: Introduction to the fundamental solution - p. 225 75. 8.2: The fundamental solution of the Laplace equation - p. 228 76. 8.3: Green's functions for the Laplace operator - p. 233 77. 8.4: The fundamental solution of the heat equation - p. 243 78. 8.5: Green's function of the heat equation - p. 252 79. 8.6: Finite-time blowup and extinction - p. 255 80. 8.7: Notes and remarks - p. 265 81. 8.8: Exercises - p. 265 82. References - p. 267 83. Chapter 9 Systems of first-order partial differential equations - p. 269 84. 9.1: Some first-order PDE models in physical sciences - p. 269 85. 9.2: The characteristic method - p. 272 86. 9.3: Well-posedness for a system of first-order PDEs - p. 276 87. 9.4: The solution of Maxwell's equations - p. 278 88. 9.5: The conservation law and shock waves - p. 282 89. 9.6: A class of semilinear wave equations - p. 289 90. 9.7: Remarks and notes - p. 296 91. 9.8: Exercises - p. 296 92. References - p. 298 93. Appendix A Some essential results in ordinary differential equations - p. 299 94. Appendix B Sobolev spaces - p. 305 95. References - p. 309 96. Index - p. 311
