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1 |
chapter one. Steady state. Heat flux. Total heat energy |
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2 |
chapter 2.4,2.5. separation of variables from selected PDE’s. Finding eigevalues for different homogeneous B.C. Heat PDE in 1D with initial conditions, homogeneous B.C. Total heat energy |
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3 |
chapter 2.5 Laplace on rectangle Laplace on quarter circle Laplace inside circular annulus backward heat PDE is not well posed. Drag force zero for uniform flow past cylinder Circulation around cylinder |
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4 |
chapters 2.5,3.2, 3.4 Fourier series, even and odd extensions Heat PDE 1D, source with homogenous B.C. |
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5 |
chapters 3.5,3.6,4.2,4.4 Fourier series, even and odd extensions Heat PDE 1D, source with homogenous B.C. Find Fourier series for \(x^m\) , Complex Fourier series Derive vibrating string wave equation. Wave equation with damping. Derive conservation of energy for vibrating string. |
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6 |
chapter 5.3,5.5 Wave equation Sturm-Liouville DE, more eigenvalue Sturm-Liouville, self adjoint Show that eigenfunctions are orthogonal. More S-L problems |
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7 |
chapter 5.6,5.9 Rayleigh quotient to find upper bound on lowest eigenvalue for S-L ODE Show eigenvalue is positive for S-L Estimating large eigenvalues for S-L with different boundary conditions Sketch eigenfunctions for \(y''+\lambda (1+x)y=0\) |
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8 |
chapter 5.10, 7.3, 7.4 How many terms needed for Fourier series for \(f(x)=1\)? Find formula for infinite series using Parseval’s equality. More on Parseval’s equality Solve Wave equation with homogeneous B.C. Solve Laplace in 3D, seperation of variables. Show that \(\lambda \ge 0\) using Rayleigh quotient. Derive Green formula |
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9 |
Chapter 8.2,8.3,8.4,8.5 Heat PDE 1D with zero source and non-homogeneous BC. Using \(u=v+u_E\) where \(u_E\) is equilibrium solution. (Use \(u_E\) if source is zero. Heat PDE 1D with source and non-homogeneous BC. Using \(u=v+u_r\) where \(u_r\) is reference solution (only needs to satisfy BC) since source is not zero. Solve heat PDE inside circle. No source, non-homogeneous BC, use \(u_E\). Solve heat PDE in 1D with time dependent \(k\) Solve heat PDE inside circle. Source, homogeneous BC. Solve heat PDE 1D. Source and non-homogeneous BC. use \(u_r\) Solve heat PDE 1D. Source and non-homogeneous BC without using \(u_r\). Solve wave equation, 1D with source and homogeneous BC. Solve wave equation, 2D membrane. With source and fixed boundaries. |
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10 |
Hand problems, not from text Solve ODE’s using two sided Green function with different boundary conditions |
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11 |
Hand problems, not from text Solve Laplace using method of images. Different boundary conditions |
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12 |
chapter 12.2 Wave equation in 1D, such as \(\frac{\partial y}{\partial t} - 3 \frac{\partial y}{\partial x} = 0\) using method of charaterstics. More wave in 1D with source. \(\frac{\partial y}{\partial t} + c \frac{\partial y}{\partial x} = e^{2 x}\) |
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