Differential and Integral EquationsDifferential and integral equations involve important mathematical techniques, and as such will be encountered by mathematicians, and physical and social scientists, in their undergraduate courses. This text provides a clear, comprehensive guide to first- and second-order ordinary and partial differential equations, whilst introducing important and useful basic material on integral equations. Readers will encounter detailed discussion of the wave, heat and Laplace equations, of Green's functions and their application to the Sturm-Liouville equation, and how to use series solutions, transform methods and phase-plane analysis. The calculus of variations will take them further into the world of applied analysis. Providing a wealth of techniques, but yet satisfying the needs of the pure mathematician, and with numerous carefully worked examples and exercises, the text is ideal for any undergraduate with basic calculus to gain a thorough grounding in 'analysis for applications'. |
Contents
0 Some Preliminaries | 1 |
1 Integral Equations and Picards Method | 5 |
2 Existence and Uniqueness | 19 |
3 The Homogeneous Linear Equation and Wronskians | 33 |
4 The NonHomogeneous Linear Equation | 41 |
5 FirstOrder Partial Differential Equations | 59 |
6 SecondOrder Partial Differential Equations | 85 |
7 The Diffusion and Wave Equations and the Equation of Laplace | 115 |
10 Iterative Methods and Neumann Series | 181 |
11 The Calculus of Variations | 189 |
12 The SturmLiouville Equation | 225 |
13 Series Solutions | 243 |
14 Transform Methods | 287 |
15 PhasePlane Analysis | 327 |
the solution of some elementary ordinary differential equations | 353 |
Bibliography | 363 |
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Common terms and phrases
boundary conditions boundary curve canonical form Chapter coefficients complete solution continuous function continuously differentiable continuously differentiable function corresponding critical point deduce defined derivatives determined discussion domain dy dx eigenfunctions eigenvalues Euler equation Example Exercise 12 exist extended power series finite Fourier transform Fredholm Alternative function f given Green’s function hence identically zero indicial equation initial conditions integral equation kernel Laplace transform Laplace’s equation Lemma linear equation linearly independent linearly independent solutions method non-homogeneous non-zero notation Note ordinary differential equations partial differential equation phase-diagram phase-paths polynomials positive constant positive integer power series problem consisting proof Proposition radius of convergence reader real constants real number real-valued function regular singular point satisfies second-order series solutions ſº solution surface solve stationary value substitution Suppose twice continuously differentiable unique solution Wronskian y1 and y2