A Second Course in Elementary Differential Equations
Published on 10. May 2014
272 pages
978-1-4832-7660-1 (ISBN)
Description
A Second Course in Elementary Differential Equations deals with norms, metric spaces, completeness, inner products, and an asymptotic behavior in a natural setting for solving problems in differential equations. The book reviews linear algebra, constant coefficient case, repeated eigenvalues, and the employment of the Putzer algorithm for nondiagonalizable coefficient matrix. The text describes, in geometrical and in an intuitive approach, Liapunov stability, qualitative behavior, the phase plane concepts, polar coordinate techniques, limit cycles, the Poincaré-Bendixson theorem. The book explores, in an analytical procedure, the existence and uniqueness theorems, metric spaces, operators, contraction mapping theorem, and initial value problems. The contraction mapping theorem concerns operators that map a given metric space into itself, in which, where an element of the metric space M, an operator merely associates with it a unique element of M. The text also tackles inner products, orthogonality, bifurcation, as well as linear boundary value problems, (particularly the Sturm-Liouville problem). The book is intended for mathematics or physics students engaged in ordinary differential equations, and for biologists, engineers, economists, or chemists who need to master the prerequisites for a graduate course in mathematics.
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Paul Waltman
A Second Course in Elementary Differential Equations
Book
10/1997
Academic Press
€40.31
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Content
Preface1 Systems of Linear Differential Equations 1. Introduction 2. Some Elementary Matrix Algebra 3. The Structure of Solutions of Homogeneous Linear Systems 4. Matrix Analysis and the Matrix Exponential 5. The Constant Coefficient Case: Real and Distinct Eigenvalues 6. The Constant Coefficient Case: Complex and Distinct Eigenvalues 7. The Constant Coefficient Case: The Putzer Algorithm 8. General Linear Systems 9. Some Elementary Stability Considerations 10. Periodic Coefficients 11. Scalar Equations 12. An Application: Coupled Oscillators2 Two-Dimensional Autonomous Systems 1. Introduction 2. The Phase Plane 3. Critical Points of Some Special Linear Systems 4. Critical Points of General TWo-Dimensional Linear Systems 5. Behavior of Nonlinear TWo-Dimensional Systems Near a Critical Point 6. Elementary Liapunov Stability Theory 7. Limit Cycles and the Poincaré-Bendixson Theorem 8. An Example: Lotka-Volterra Competition 9. An Example: The Simple Pendulum3 Existence Theory 1. Introduction 2. Preliminaries 3. The Contraction Mapping Theorem 4. The Initial Value Problem for One Scalar Differential Equation 5. The Initial Value Problem for Systems of Differential Equations 6. An Existence Theorem for a Boundary Value Problem4 Boundary Value Problems 1. Introduction 2. Linear Boundary Value Problems 3. Oscillation and Comparison Theorems 4. Sturm-Liouville Problems 5. The Existence of Eigenvalues for Sturm-Liouville Problems 6. Tw¿ Properties of Eigenfunctions 7. An Alternate Formulation-Integral Equations 8. Eigenfunction Expansions 9. The Inhomogeneous Sturm-Liouville Problem 10. Some Standard Applications of Sturm-Liouville Theory 11. Nonlinear Boundary Value ProblemsIndex
