Partial Differential Equations with MATLAB
6 An Introduction to Partial Differential Equations with MATLAB©R c) Isu=x+ya solution? d) Find all solutions of the formu=ax+by ...
Introduction 7 Since u=f(y) (1.3) represents all possible solutions of (1.1), we call (1.2) thegeneral solutionof (1.1). So, whe ...
8 An Introduction to Partial Differential Equations with MATLAB©R and our general solution is u=ysinx+f 1 (x)+g(y). Finally, sin ...
Introduction 9 Exercises 1.2 Find the general solution of each PDE. The solutionuis a function of the variables which appear, un ...
10 An Introduction to Partial Differential Equations with MATLAB©R 1.3 Initial and Boundary Conditions In some of the exercises ...
Introduction 11 Alternatively, the right end may be insulated. We will see that, mathemati- cally, this means that ux(L, t)=0,t& ...
12 An Introduction to Partial Differential Equations with MATLAB©R however, for the ODE problems in Exercises 1–4. Each of these ...
Introduction 13 and that, ify 1 andy 2 are any functions in the domain ofL,then L[y 1 +y 2 ]=L[y 1 ]+L[y 2 ]. We use the idea of ...
14 An Introduction to Partial Differential Equations with MATLAB©R We will prove, in Exercise 8, thatLis linear if and only if L ...
Introduction 15 In practice, the things that make ODEs nonlinear also make PDEs nonlin- ear, for example, powers ofuand its deri ...
16 An Introduction to Partial Differential Equations with MATLAB©R Prove that the operatorL[u] is linear if and only if it sati ...
Introduction 17 Theorem 1.1Ifu 1 ,u 2 ,...,unare solutions of the linear, homogeneous PDE L[u]=0, then so is any linear combinat ...
18 An Introduction to Partial Differential Equations with MATLAB©R The heat equation in two space variables,ut=α^2 (uxx+uyy) La ...
Introduction 19 1.6 Separation of Variables for Linear, Homogeneous PDEs In the mid-1700s, Daniel Bernoulli and, later, Jean le ...
20 An Introduction to Partial Differential Equations with MATLAB©R that is, we have managed toseparatethe variablexfrom the vari ...
Introduction 21 Again, dividing both sides byugives us T′ T = X′′ X = constant. For the sake of convenience (we’ll see why later ...
22 An Introduction to Partial Differential Equations with MATLAB©R Example 4Separate the PDE (inu(x, y, z)), ux− 2 uyy+3uz=0. We ...
Introduction 23 Exercises 1.6 In Exercises 1–21, separate the PDE into a system of ODEs. 3ux− 2 uy=0 5ux+4uy− 2 u=0 3.y^2 ux+x ...
24 An Introduction to Partial Differential Equations with MATLAB©R 5ux+4uy− 2 u=0 24.y^2 ux+x^2 uy=0 25.uxx−uy+u=0 The wave e ...
Introduction 25 (The last two boundary conditions are encountered in connection with the Euler–Bernoulli beam PDE, for example.) ...
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