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CHAPTER 11 Partial Differential Equations | Change ‘The Fourier transform method has been included in the section on Fourier integrals for heat problems (Sec. 11.6). SECTION 11.1 Basic Concepts, page 583 Purpose. To familiarize the student with the following: Concept of solution, verification of solutions Superposition principle for homogeneous linear equations Equations solvable by methods for ordinary differential equations i SOLUTIONS TO PROBLEM SET 11.1, page 584 ' 16. 4 = A(y) cos 3x + B(y) sin 3x + 18. u,fu = —2y, Inu = —y? + B®), uw = e(x) exp (—y?) 20. = cy(x)e¥ + co(xe7# 22. ty = 9 ay = 4, q = Bade, u = [q dy = cade + Hw) 24, By the chain rule, (A) Man ~ Ry = VErta + Zoe) — XErty + Zo) = 0. Now r= (x? + y?)¥?, 1, = 4G? + y)-Y? 2x = xr, ry = ylr, so that We ~ xr = 0 in (A) and (A) gives zy = 0. SECTION 11.2. Modeling: Vibrating String, Wave Equation, Page 585 Purpose. A careful derivation of the one-dimensional wave equation (more careful than in most other texts, where some of the essential physical assumptions are usually missing). Short Courses. One may perhaps omit the derivation and Just state the wave equation and mention of what c? is composed. i SECTION 11.3. Separation of Variables. Use of Fourier Series, page 587 Purpose. This first section in which we solve a “big” problem has several purposes: 1, To familiarize the student with the wave equation and with the typical initial and boundary conditions that physically meaningful solutions must satisfy. 2. To explain and apply the important method of separation of variables, by which the partial differential equation is reduced to ordinary differential equations. i 3. To show how Fourier series help to get the final answer, thus seeing the reward of our great and long effort in Chap. 10. 4. To discuss the eigenfunctions of the problem, the basic building blocks of the solu- tion, which lead to a deeper understanding of the whole problem. 161 docsity.com 
