This serves as a text for a graduate level course in the theory of ordinary differential equations, written from a dynamical systems point of view. It contains both theory and applications, with the applications interwoven with the theory throughout the text. The author also links ordinary differential equations with advanced mathematical topics such as differential geometry, Lie group theory, analysis in infinite-dimensional spaces and even abstract algebra. This textbook is a unique blend of the theory of differential equations and their exciting application to "real world" problems. First, and foremost, it is a rigorous study of ordinary differential equations and can be fully understood by anyone who has completed one year of calculus. However, in addition to the traditional applications, it also contains many exciting "real life" problems. These applications are completely self-contained. Written in a straightforward and easily accessible style, this volume is suitable as a textbook for advanced undergraduate or first-year graduate students in mathematics, physical sciences, and engineering. The aim is to provide students with a strong background in the theories of Ordinary Differential Equations, Dynamical Systems and Boundary Value Problems, including regular and singular perturbations. It is also a valuable resource for researchers. This volume presents an abundance of examples in physical and biological sciences, and engineering to illustrate the applications of the theorems in the text. Readers are introduced to some important theorems in Nonlinear Analysis, for example, Brouwer fixed point theorem and fundamental theorem of algebras. A chapter on Monotone Dynamical Systems takes care of the new developments in Ordinary Differential Equations and Dynamical Systems. An introduction to Hamiltonian Systems is included to enhance and complete its coverage on Ordinary Differential Equations with applications in Mathematical Biology and Classical Mechanics. Ordinary Differential Equations with Applications blends the theory and practical applications of Ordinary Differential Equations (ODEs) with real-world examples. It covers fundamental ODE concepts, from first-order equations to more advanced topics like the Laplace and Mellin transforms, Fourier series, and power series solutions. Designed for undergraduate and postgraduate students in mathematics, physics, engineering, and other fields, it is also a valuable resource for professionals. The book addresses various applications in biology, economics, chemistry, and medicine.
Carlos Enrique Frasser earned his undergraduate degree from the State University of New York at Albany and Ph.D. in mathematics at the University of Wisconsin. After one year as a visiting assistant professor at the University of Missouri-Columbia, he joined the faculty and has been a Professor of Mathematics. He works in the areas of dynamical systems and applied mathematics. He has published over 70 research articles and two books. His recent research interests are in the qualitative theory of ordinary and partial differential equations, applications of differential equations to mathematical models in science and engineering, and the development of numerical methods to obtain viable predictions from physically realistic model problems.
| Preface Chapter 1. Introduction to Differential Equations Definition and Classification of Differential Equations Order and Degree of Differential Equations Ordinary vs. Partial Differential Equations Initial Value and Boundary Value Problems Chapter 2. First-Order Differential Equations Separable Differential Equations Linear First-Order Equations Exact Differential Equations Integrating Factors Bernoulli and Riccati Equations Chapter 3. Second-Order Linear Differential Equations Homogeneous Linear Equations with Constant Coefficients Characteristic Equations and General Solutions Particular Solutions Using Undetermined Coefficients Variation of Parameters Mechanical Vibrations and Damped Systems Chapter 4. Higher-Order Linear Differential Equations Linear Independence and the Wronskian An Nth Order Homogeneous Linear Equations Nonhomogeneous Equations with Constant Coefficients Method of Annihilators Cauchy-Euler Equations Applications to Beam Deflection and Circuits Chapter 5. Systems of Differential Equations Introduction to Systems of ODEs Matrix Methods and Eigenvalues Linear Systems with Constant Coefficients Diagonalization and Matrix Exponentials Chapter 6. Series Solutions of ODEs Power Series Method Radius and Interval of Convergence Series Solutions Near Ordinary Points Frobenius Method |
Chapter 7. Laplace Transforms Definition and Properties of Laplace Transforms Inverse Laplace Transforms Solving Initial Value Problems Convolution Theorem Chapter 8. Numerical Methods for ODEs Euler’s Method Improved Euler and Heun’s Methods Runge-Kutta Methods Multistep Methods (e.g., Adams-Bashforth) Chapter 9. Qualitative Analysis of ODEs Phase Portraits and Direction Fields Critical Points and Linearization Stability of Equilibrium Points Lyapunov’s Direct Method Chapter 10. Special Functions in ODEs Gamma and Beta Functions Bessel Functions of the First and Second Kind Legendre Polynomials Hermite and Chebyshev Polynomials Chapter 11. Transform Methods Beyond Laplace Fourier Series Basics Fourier Transforms Applications to Boundary Value Problems Wavelet Transforms: Localized BoundaryValue Analysis Sine and Cosine Transforms Applications in Signal Processing Bibliography |
$280.00