Many real-world issues may be represented mathematically using differential equations. The solution of the differential equation then provides the answer to the underlying problem. The solution to the differential equation is therefore crucial to the real-world problem. Fluid dynamics, heat transfer, boundary layer flow, elasticity, quantum mechanics, and electromagnetic theory are just a few of the fields of applied mathematics where partial differential equations are used. Only a few of these equations can be solved using analytical approaches, which are also complicated by the fact that they necessitate the application of complex mathematical techniques. The method of finite differences is the most often utilised of all the numerical methods for solving partial differential equations. Although there exist various numerical methods for solving PDEs, scholars have long strived to develop simple and efficient approaches.
This book is an attempt to bring together the disparate knowledge on numerical solution of partial differential equations. The chapters in this book cover a wide spectrum of scientific disciplines such as two-point boundary value problems, system of algebraic equations, heat equation and wave equation. It discusses thoroughly the various basic and fundamental concepts of this subject. It also deals with maximum principle, poisson's equation, orthogonality, reaction-diffusion equations and fourier series. This book presents this complex subject in the most comprehensible and easy to understand language. It explores all the important aspects of the area in the present day scenario. This book is meant for students who are looking for an elaborate reference book on numerical solution of partial differential equations.
1. Introduction to Differential Equations 1
2. Concepts 1
3. The Solution and Its Properties 6
4. An Ordinary Differential Equation 10
5. A Numerical Method 12
6. Cauchy Problems 19
7. First-Order Homogeneous Equations 25
8. First-Order Nonhomogeneous Equations 27
9. The Wave Equation 30
10. Conclusion 33
References 33
2. Two-Point Boundary Value Problems 37
1. Introduction to Boundary Value Problems 37
2. Poisson's Equation in One Dimension 41
3. Green's Function 42
4. A Finite Difference Approximation 52
5. Conclusion 54
References 55
3. A System of Algebraic Equations 57
1. An Introduction System of Algebraic Equations 57
2. Gaussian Elimination for Tridiagonal Linear Systems 76
3. Diagonal Dominant Matrices 80
4. Positive Definite Matrices 83
5. Continuous and Discrete Solutions 88
6. Uniqueness Theorem 92
8. Eigenvalue Problems 94
9. Conclusion 100
References 100
4. The Heat Equation 103
1. A Brief Overview 103
2. Heat on an insulated wire 111
3. Fourier Coefficients 120
4. Conclusion 133
References 133
5. The Wave Equation 136
1. Introduction 136
2. Derivation of the Wave Equation 137
3. Scalar wave equation in three space dimensions 145
4. Scalar wave equation in two space dimensions 148
5. Separation of Variables 152
6. Inhomogeneous wave equation in one dimension 156
7. Conclusion 158
References 159
6. The Maximum Principle 162
1. An Introduction to maximum principles 162
2. A Two-Point Boundary Value Problem 163
3. Linear Explicit Finite Difference Scheme 164
4. The Implicit Finite Difference Scheme 164
5. The Nonlinear Heat Equation 165
6. An Explicit Finite Difference Scheme 173
7. Harmonic Functions 174
8. Conclusion 176
References 177
7. Poisson's Equation in Two Space Dimensions 180
1. Introduction 180
2. Poisson equation in Rectangular Domains 181
3. The Disc 182
4. AWedge 183
5. Applications of the Divergence Theorem 184
5. Stating the Divergence Theorem 186
6. Using the Divergence Theorem 188
7. A Finite Difference Approximation 189
8. The Five-Point Stencil 189
9. An Error Estimate 191
10. Gaussian Elimination for General Systems 192
11. Upper Triangular Systems 194
12. General Matrix Systems 196
13. Banded Systems 197
15. Conclusion 198
References 199
8. Orthogonality and General Fourier Series 201
1. Introduction to Fourier Series 201
2. The Full Fourier Series 204
3. Even and Odd functions 207
4. Differentiation of Fourier Series 213
5. Sturm-Liouville Problems 217
6. The Mean Square Distance 217
7. A Poincaré Inequality 218
8. Conclusion 219
References 220
9. Convergence of Fourier Series 223
1. Introduction of Convergence of Fourier Series 223
2. Pointwise convergence 225
3. Uniform convergence 226
4. Absolute convergence 226
5. Norm convergence 227
6. Conclusion 233
References 233
10 Reaction-Diffusion Equations 238
1. Introduction to Reaction-Diffusion Equations 238
2. Applications of Fisher’s Equation 243
3. Conclusion 249
References 249
Ayako Singer pursued her Ph.D. in Applied Mathematics from Institute of Technology, Aichi. She serves as an Adjunct Professor at University College, Changwon. She has keen interest in the areas of coordinate geometry, probability theory, differential calculus, and advanced calculus. Singer has contributed several research papers and articles on differential equations, algebraic equations and discrete mathematics. She has attended numerous conferences and symposiums and has presented 15 papers on differential equations and mathematical physics.
