Advanced Mathematical Methods for Scientists and Engineers I: Asymptotic Methods and Perturbation TheorySpringer Science & Business Media, 2013. 3. 9. - 593ÆäÀÌÁö The triumphant vindication of bold theories-are these not the pride and justification of our life's work? -Sherlock Holmes, The Valley of Fear Sir Arthur Conan Doyle The main purpose of our book is to present and explain mathematical methods for obtaining approximate analytical solutions to differential and difference equations that cannot be solved exactly. Our objective is to help young and also establiShed scientists and engineers to build the skills necessary to analyze equations that they encounter in their work. Our presentation is aimed at developing the insights and techniques that are most useful for attacking new problems. We do not emphasize special methods and tricks which work only for the classical transcendental functions; we do not dwell on equations whose exact solutions are known. The mathematical methods discussed in this book are known collectively as asymptotic and perturbative analysis. These are the most useful and powerful methods for finding approximate solutions to equations, but they are difficult to justify rigorously. Thus, we concentrate on the most fruitful aspect of applied analysis; namely, obtaining the answer. We stress care but not rigor. To explain our approach, we compare our goals with those of a freshman calculus course. A beginning calculus course is considered successful if the students have learned how to solve problems using calculus. |
¸ñÂ÷
5 | |
9 Differential Equations in the Complex Plane | 29 |
Difference Equations | 36 |
5 Nonlinear Difference Equations | 53 |
PART II | 60 |
2 Local Behavior Near Ordinary Points of Homogeneous Linear | 66 |
4 Local Behavior at Irregular Singular Points of Homogeneous | 76 |
6 Local Analysis of Inhomogeneous Linear Equations | 103 |
Problems for Chapter 4 | 196 |
Approximate Solution of Difference Equations | 205 |
Asymptotic Expansion of Integrals | 247 |
5 Mathematical Structure of Perturbative Eigenvalue Problems | 350 |
Summation of Series | 368 |
2 Summation of Divergent Series | 379 |
5 Convergence of Padé Approximants | 400 |
PART III | 417 |
8 Asymptotic Series | 118 |
Problems for Chapter 3 | 136 |
Approximate Solution of Nonlinear Differential Equations | 146 |
EF | 157 |
Problems for Chapter 9 | 479 |
582 | |
±âŸ ÃâÆǺ» - ¸ðµÎ º¸±â
Advanced Mathematical Methods for Scientists and Engineers Carl M. Bender,Steven A. Orszag ªÀº ¹ßÃé¹® º¸±â - 1978 |
Advanced Mathematical Methods for Scientists and Engineers Carl M. Bender,Steven A. Orszag ªÀº ¹ßÃé¹® º¸±â - 1978 |
ÀÚÁÖ ³ª¿À´Â ´Ü¾î ¹× ±¸¹®
a©û a©ü an+1 analysis analytic approximation to y(x asymptotic behavior asymptotic expansion asymptotic matching asymptotic relation asymptotic series B©û behavior of solutions behavior of y(x boundary condition boundary layer boundary-layer theory boundary-value problem branch points c©û c©ü coefficients compute constant continued fraction contour critical point derive determined difference equation differential equation Example exponentially finite first-order formula gives higher-order initial conditions irregular singular point leading behavior leading-order linear local analysis method multiple-scale nonlinear obtain outer solution Padé approximants Padé sequence perturbation series perturbation theory plane plot poles polynomial power series Prob radius of convergence region relative error result saddle point satisfies second-order Shanks transformation Show solve Stieltjes function Substituting Taylor series tion trajectories uniform approximation valid values WKB approximation WKB theory y©û y©û(x Yo(X yout(x