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Full Description
This book illustrates how MAPLE™ can be used to supplement a standard, elementary text in ordinary and partial differential equation. The authors are firm believers in the teaching of mathematics as an experimental science where the student does numerous calculations and then synthesizes these experiments into a general theory. The goal of the book is to teach the students enough about the computer algebra system MAPLE™ so that it can be used in an investigative way. This book was developed through ten years of instruction in the differential equations course.
Contents
Preface. 1. Introduction to the Maple DEtools. 1.1. Analytical Solutions and their Plotting. 1.2. Direction Fields and Integral Curves. 1.3. Computer Lab. 1.4. Supplementary Maple Programs. 2. First-order Differential Equations. 2.1. Linear differential equations. 2.2. Project: Mixing problems. 2.3. Separable differential equations. 2.4. Exact Questions. 3. Numerical Methods for First Order Equations. 3.1. Picard's Iteration Method and Semi-batch Reactor. 3.2. An Existence and Uniqueness Theorem. 3.3. Picard Iteration Method. 3.4. Computer Lab. 3.5. Numerical Procedures and Fermentation Kinetics. 3.6. The Euler Method. 3.7. Higher-Order Methods. 3.8. Maple Procedures. 3.9. Computer Lab. 3.10. Supplementary Maple Programs. 4. Differential Equations with Constant Coefficients. 4.1. Second order equations with constant coefficients. 4.2. Variation of Parameters. 4.3. The Method of Undetermined Coefficients. 4.4. Higher order, homogeneous equations. 4.5. Nonhomogeneous Linear Equations. 5. Applications of Second Order Linear Equations. 5.1. Simple Harmonic Motion. 5.2. General Solutions. 5.3. Method of Undetermined Coefficients. 5.4. Additional Useful Commands. 5.5. Computer Lab. 5.6. Supplementary Maple Programs. 5.7. Particular Solutions. 5.8. Computer Lab. 5.9. Supplementary Maple Programs. 6. Two-Point Boundary Value Problems, Catalytic Reactors and Boundary-Layer Phenomena. 6.1. Analytical Solutions. 6.2. Finite-Difference Methods. 6.3. Computer Lab. 6.4. Supplementary Maple Programs. 7. Eigenvalue Problems. 7.1. Sturm-Liouville Problems. 7.2 Numerical Approximations. 7.3. The Newton-Raphson Method. 7.4. Computer Lab. 7.5. Supplementary Mapple Programs. 8. Power Series Methods for Solving Differential Equations. 8.1. Nonlinear Differential Equations. 8.2. Regular-Singular Points. 8.3. Programs for finding solutions. 8.4. Projects. 9. Nonlinear Autonomous Systems. 9.1. The Taylor Series Method. 9.2. The Phase Plane. 9.3. Linear Systems. 9.4. Useful Maple Commands. 9.5. Computer Lab. 9.6. Supplementary Maple Programs. 10. Integral Transforms. 10.1 The Laplace Transform of Elementary Functions. 10.2. Solving Differential Equations with the Laplace Transform. 10.3. Fourier Transforms. 11. Partial Differential Equations. 11.1. Elementary Methods. 11.2. The First Order Partial Differential Equation. 11.3. The Heat Equation. 11.4. The Vibrating String. 11.5 The Laplace Equation. 12. Transmutations. 12.1. The method of ascent. 12.2. Orthogonal systems of functions. 12.3. Acoustic propagation. Bibliography. Index.
