1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers
48 Chapter 0 Ordinary Differential Equations has one and only one solution, unless there is a nontrivial solution of d^2 u dx^2 ...
0.5 Green’s Functions 49 The general solution of the corresponding homogeneous equation isu(x)= c 1 +c 2 ln(x).Thus,wewouldchoos ...
50 Chapter 0 Ordinary Differential Equations d (^2) u dx^2 −γ^2 u=f(x), −∞<x<∞, u(x)bounded asx→±∞. Use the Green’s fun ...
Miscellaneous Exercises 51 14.Show that the boundary value problem d^2 u dx^2 +λ^2 u=f(x), 0 <x<a, u( 0 )= 0 , u(a)= 0 , w ...
52 Chapter 0 Ordinary Differential Equations 6. 1 r d dr ( r du dr ) =0, a<r<b, u(a)=T 0 , u(b)=T 1. 7.^1 ρ^2 d dρ ( ρ^2 d ...
Miscellaneous Exercises 53 15.Solve foru(x).Notetheinterval. d^4 u dx^4 + k EI u=w, 0 <x<∞ (wis constant), u( 0 )= 0 , d^2 ...
54 Chapter 0 Ordinary Differential Equations 21.Solve the differential equation d^2 u dx^2 =p^2 u, 0 <x<a, subject to the ...
Miscellaneous Exercises 55 that follows, which can be done in closed form. d^2 u dx^2 =γ (^2) u (^4) , 0 <x, u( 0 )=U, xlim→∞ ...
56 Chapter 0 Ordinary Differential Equations whereDis thediffusion constant; and (2) when the sulphur dioxide reacts with water, ...
Miscellaneous Exercises 57 Show that this relation is a differential equation and solve it. (Call the constantp^2 .) Prove that ...
This page intentionally left blank ...
Fourier Series and Integrals CHAPTER 1 1.1 Periodic Functions and Fourier Series Afunctionfis said to beperiodic with period p&g ...
60 Chapter 1 Fourier Series and Integrals Figure 1 A periodic function of periodp. functions have the common period 2π, although ...
Chapter 1 Fourier Series and Integrals 61 ∫π −π sin(nx)dx= 0 ∫π −π cos(nx)dx= { 0 , n= 0 2 π, n= 0 ∫π −π sin(nx)cos(mx)dx= 0 ∫π ...
62 Chapter 1 Fourier Series and Integrals We can now summarize our results. In order for the proposed equality f(x)=a 0 + ∑∞ n= ...
Chapter 1 Fourier Series and Integrals 63 Figure 2 f(x)=x,−π<x<π,fperiodic with period 2π. Thus, for this function, we hav ...
64 Chapter 1 Fourier Series and Integrals b. f(x)= { 0 , − 1 <x≤0, x, 0 <x<1, f(x+^2 )=f(x); c. f(x)= { 0 , −π<x≤0, ...
1.2 Arbitrary Period and Half-Range Expansions 65 case, the coefficients are a 0 =^1 2 a ∫a −a f(x)dx, an=^1 a ∫a −a f(x)cos (nπ ...
66 Chapter 1 Fourier Series and Integrals of period 2a, by using the following definitions: ̄f(x)=f(x), −a<x<a, ̄f(x)=f(x+ ...
1.2 Arbitrary Period and Half-Range Expansions 67 Figure 3 f(x)=x,− 1 <x<1,fperiodic with period 2. Definition Afunctiong( ...
«
1
2
3
4
5
6
7
8
9
10
»
Free download pdf