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on 12-Dec-2023 (Tue)

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Annotation 7604680920332

Using Fourier series to solve 2nd linear inhomogeneous ODE particular solution

For a 2nd order linear inhomogeneous ODE

\(\frac{d^{2}y}{dx^{2}}+p\frac{dy}{dx}+qy(x)=f(x)\)

where \(f(x)\) is a periodic function that given the period \(L_{0}\).
To find the ODE's particular integral,
1. Calculate the Fourier series of the function \(f(x)\)
2. Try ansatz and using the period \(L_{0}\): \(\displaystyle y(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L_{0}}\right)+b_{n}\sin\left(\frac{n\pi x}{L_{0}}\right)\).
3. Yield the derivatives of \(y(x)\) and plug them into the original ODE.
4. Comparing coefficients in front of 1, cosine, sine, respectively, write out the equation system about the ODE.
5. Solve the equation system and write out the particular integral.
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Flashcard 7604682755340

Question

For a 2nd order linear inhomogeneous ODE

\(\frac{d^{2}y}{dx^{2}}+p\frac{dy}{dx}+qy(x)=f(x)\)

where \(f(x)\) is a periodic function that is given the period \(L_{0}\).
Given \(p, q, f(x)\), to find the ODE's particular integral, [Using Fourier series to solve 2nd linear inhomogeneous ODE particular solution]

Answer

1. Calculate the Fourier series of the function \(f(x)\)
2. Try ansatz and using the period \(L_{0}\): \(\displaystyle y(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L_{0}}\right)+b_{n}\sin\left(\frac{n\pi x}{L_{0}}\right)\).
3. Yield the derivatives of \(y(x)\) and plug them into the original ODE.
4. Comparing coefficients in front of 1, cosine, sine, respectively, write out the equation system about the ODE.
5. Solve the equation system and write out the particular integral.

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Using Fourier series to solve 2nd linear inhomogeneous ODE particular solution
order linear inhomogeneous ODE \(\frac{d^{2}y}{dx^{2}}+p\frac{dy}{dx}+qy(x)=f(x)\) where \(f(x)\) is a periodic function that given the period \(L_{0}\). To find the ODE's particular integral, <span>1. Calculate the Fourier series of the function \(f(x)\) 2. Try ansatz and using the period \(L_{0}\): \(\displaystyle y(x)=\frac{a_{0}}{2}+\sum\limits^{\infty}_{n=1}a_{n}\cos\left(\frac{n\pi x}{L_{0}}\right)+b_{n}\sin\left(\frac{n\pi x}{L_{0}}\right)\). 3. Yield the derivatives of \(y(x)\) and plug them into the original ODE. 4. Comparing coefficients in front of 1, cosine, sine, respectively, write out the equation system about the ODE. 5. Solve the equation system and write out the particular integral. <span>






Annotation 7604685376780

Main ingredients of a PDE
Four things form the basic structure of any PDE problem:
- TF: Target function(Function of interests \(\phi(x,t)\))
- DOV: Domain of Validity, such as \(x_{1}<x<x_{2}\) and \(t_{1}<t<t_{2}\).
- PDE: Partial Differential Equation
- BC: Boundary condition
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Flashcard 7604687998220

Question
Four things form the basic structure of any PDE problem: [...]
Answer

- TF: Target function(Function of interests \(\phi(x,t)\))
- DOV: Domain of Validity, such as \(x_{1}<x<x_{2}\) and \(t_{1}<t<t_{2}\).
- PDE: Partial Differential Equation
- BC: Boundary condition

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Main ingredients of a PDE
Four things form the basic structure of any PDE problem: - TF: Target function(Function of interests \(\phi(x,t)\)) - DOV: Domain of Validity, such as \(x_{1}<x<x_{2}\) and \(t_{1}<t<t_{2}\). - PDE: Partial Differential Equation - BC: Boundary condition






Annotation 7604700056844

General Form for 2nd linear homogeneous PDE

The general form of \(2^{nd}\) order linear homogeneous PDE is

\(\displaystyle A(x)\phi_{xx}+B(x)\phi_{xy}+C(x)\phi_{yy}+D(x)\phi_{x}+E(x)\phi_{y}+F(x)\phi=0\)

where \(A(x),B(x),C(x),D(x),E(x),F(x)\) are the function of \(x\), and \(\phi_{xx}\) stands for \(\displaystyle\frac{\partial^{2}\phi}{\partial x^{2}}\), \(\phi_{x}\) stands for \(\displaystyle\frac{\partial\phi}{\partial x}\), etc.
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Flashcard 7604701891852

Question
The general form of \(2^{nd}\) order linear homogeneous PDE is: [...]
Answer

\(\displaystyle A(x)\phi_{xx}+B(x)\phi_{xy}+C(x)\phi_{yy}+D(x)\phi_{x}+E(x)\phi_{y}+F(x)\phi=0\)

where \(A(x),B(x),C(x),D(x),E(x),F(x)\) are the function of \(x\), and \(\phi_{xx}\) stands for \(\displaystyle\frac{\partial^{2}\phi}{\partial x^{2}}\), \(\phi_{x}\) stands for \(\displaystyle\frac{\partial\phi}{\partial x}\), etc.


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General Form for 2nd linear homogeneous PDE
The general form of \(2^{nd}\) order linear homogeneous PDE is \(\displaystyle A(x)\phi_{xx}+B(x)\phi_{xy}+C(x)\phi_{yy}+D(x)\phi_{x}+E(x)\phi_{y}+F(x)\phi=0\) where \(A(x),B(x),C(x),D(x),E(x),F(x)\) are the function of \(x\), and \(\phi_{xx}\) stands for \(\displaystyle\frac{\partial^{2}\phi}{\partial x^{2}}\), \(\phi_{x}\) stands for \(\displaystyle\frac{\partial\phi}{\partial x}\), etc.






Annotation 7604706348300

3 types of linear homogeneous PDE

From the general form of the linear homogeneous PDE, we can divide all PDE into 3 types:
1. \(\Delta\equiv B^{2}-4AC>0\Rightarrow\) Hyperbolic PDE(Wave Equation): \(\phi_{tt}=c\phi_{xx}\)
2. \(\Delta\equiv B^{2}-4AC=0\Rightarrow\) Parabolic PDE(Heat/Diffusion Equation): \(\phi_{t}=k\phi_{xx}\)
3. \(\Delta\equiv B^{2}-4AC<0\Rightarrow\) Elliptical PDE(Laplace Equation): \(\phi_{xx}+c\phi_{yy}=0\)
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Flashcard 7604708183308

Question
From the general form of the linear homogeneous PDE, we can divide all PDE into 3 types: [...]
Answer

1. \(\Delta\equiv B^{2}-4AC>0\Rightarrow\) Hyperbolic PDE(Wave Equation): \(\phi_{tt}=c\phi_{xx}\)
2. \(\Delta\equiv B^{2}-4AC=0\Rightarrow\) Parabolic PDE(Heat/Diffusion Equation): \(\phi_{t}=k\phi_{xx}\)
3. \(\Delta\equiv B^{2}-4AC<0\Rightarrow\) Elliptical PDE(Laplace Equation): \(\phi_{xx}+c\phi_{yy}=0\)

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3 types of PDE
From the general form of the linear homogeneous PDE, we can divide all PDE into 3 types: 1. \(\Delta\equiv B^{2}-4AC>0\Rightarrow\) Hyperbolic PDE(Wave Equation): \(\phi_{tt}=c\phi_{xx}\) 2. \(\Delta\equiv B^{2}-4AC=0\Rightarrow\) Parabolic PDE(Heat/Diffusion Equation): \(\phi_{t}=k\phi_{xx}\) 3. \(\Delta\equiv B^{2}-4AC<0\Rightarrow\) Elliptical PDE(Laplace Equation): \(\phi_{xx}+c\phi_{yy}=0\)