Partial Differential Equations with MATLAB
26 An Introduction to Partial Differential Equations with MATLAB©R the latter in that it includes theparameterλ. Remember that w ...
Introduction 27 Applying the left end boundary condition, we have y(0) = 0 =c 1 cosh 0 +c 2 sinh 0 =c 1. So the only solutions t ...
28 An Introduction to Partial Differential Equations with MATLAB©R As in Case 1, this forcesc 2 =0except in those cases wherekis ...
Introduction 29 Case 2: λ=0 The general solution is y=c 1 x+c 2 so that y′=c 1. Then, y′(0) =y′(3) = 0 =c 1. Therefore, the func ...
30 An Introduction to Partial Differential Equations with MATLAB©R First, note that this is a Cauchy–Euler equation, forx>0. ...
Introduction 31 Applying the boundary conditions, we have y(1) = 0 =c 1 ⇒c 1 =0 y(e)=0=c 2 sink and the latter equation forcesc ...
32 An Introduction to Partial Differential Equations with MATLAB©R Therefore (see Exercise 23),f(x) has no roots whenx>0 and ...
Introduction 33 FIGURE 1.2 MATLAB graph of the intersection of the functionsyyy===−−−kkk and yyy=tan=tan=tankkkfork>k>k> ...
34 An Introduction to Partial Differential Equations with MATLAB©R FIGURE 1.3 MATLAB graph of the first five eigenfunctions in E ...
Introduction 35 Now, if we write these equations in matrix form, [ 1 k cosksink ][ c 1 c 2 ] = [ 0 0 ] , we see thatλ=k^2 is an ...
36 An Introduction to Partial Differential Equations with MATLAB©R i) by hand ii) MATLAB:Using the MATLAB routine BVP4C a) y′′+λ ...
Introduction 37 20.Sturm Comparison Theorem:TheSturm Comparison Theorem(due to Jacques Charles Fran ̧cois Sturm, whom we’ll meet ...
38 An Introduction to Partial Differential Equations with MATLAB©R a) Show that (λ 1 −λ 2 ) ∫L 0 y^1 y^2 dx= ∫L 0 (y^1 y ′′ 2 −y ...
Introduction 39 c) Show that the eigenvalue problem y′′+λy=0, 0 <x<L, y(0)−ay′(0) =y(L)+by′(L)=0 has no negative eigenvalu ...
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Prelude to Chapter 2 In this chapter, we provide physical derivations for the three most important PDEs, the heat equation, the ...
42 An Introduction to Partial Differential Equations with MATLAB©R and 1780s, but his greatest contribution was his landmark fiv ...
2 The Big Three PDEs 2.1 Second-Order, Linear, Homogeneous PDEs with Constant Coefficients In this chapter we begin to look at t ...
44 An Introduction to Partial Differential Equations with MATLAB©R d) Show that, in each of the above, if we interchange the ind ...
The Big Three PDEs 45 0 x x+ xΔ L FIGURE 2.2 Differential element of lengthΔΔΔxxx. First, heat content will be defined so that t ...
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