Partial Differential Equation Note

Piecewise Continuous

Definition

Function $f$ is piecewise continuous if both $f(x_{k-1}^{+})$ and $f(x_k^{-})$ where $(k=1,2,\dots ,n)$, are continuous

Notations

$C_p(a,b)$: The set^[vector space] of all piecewise continuous functions on $(a,b)$

Facts

A continuous function may not be piecewise continuous

If $f$ is piecewise continuous on $(a,b)$, we can define ${\int }_a^{b}f(x)dx={\sum }_{k=1}^n{\int }_{k-1}^{k}f(x)dx$

For any two piecewise functions $f,g$. $f+g$ is piecewise continuous and $c\cdot f$ is piecewise continuous

If $f$ is piecewise continuous on $(a,b)$, then $f$ is bounded ($\forall x\in (a,b):|f(X)|\le M$) and ${\int }_a^{b}f(x)dx$, ${\int }_a^{b}f(x)\cos lxdx$, ${\int }_a^{b}f(x)\sin lxdx$ are finite, where $l\in \mathbb{R}$

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Fourier Series

Definition

Continuous Fourier Series

Sine-Cosine Form

$f(x)\sim A_0+\sum _{n=1}^{\infty }\left(A_n\cos (nx\frac{\pi }{L})+B_n\sin (nx\frac{\pi }{L})\right)$ where $L$ is half of the period of the $f(x)\in C_p(-L,L)$¹, $A_0=\frac{1}{2L}{\int }_{-L}^{L}f(x)dx$, $A_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\cos (nx\frac{\pi }{L})dx$, and $B_n=\frac{1}{L}{\int }_{-L}^{L}f(x)\sin (nx\frac{\pi }{L})dx$

Exponential Form

$f(x)\sim \sum _{n=-\infty }^{\infty }\left(c_n{e}^{inx\frac{\pi }{L}}\right)$ where $L$ is half of the period of the $f(x)\in C_p(-L,L)$¹, $c_n=\frac{1}{L}{\int }_{-L}^{L}f(x)e^{-inx\frac{\pi }{L}}dx$

Discrete Fourier Series

$x[n]=\sum _{k}X[k]e^{i2\pi \frac{k}{N}n},n\in \mathbb{Z}$ where $\frac{1}{N}$ is a fundamental frequency, and for some positive $N$, the practical range of $k$ is $[0,N-1]$, and $X[k]=\frac{1}{N}\sum _{k}x[n]e^{ik\frac{2\pi }{N}n}$

Facts

Suppose that

• $f$ is continuous on $(-\pi ,\pi )$
• $f(\pi )=f(-\pi )$
• $f^{\prime }\in C_p(-\pi ,\pi )$¹

Then ${\sum }_{n=1}^{\infty }\sqrt{B_n^2+B_n^2}$ converges where $A_n,B_n$ are the coefficients of Fourier series It is stronger than Bessel’s Inequality

Suppose that

• $f$ is continuous on $[-\pi ,\pi ]$
• $f(\pi )=f(-\pi )$
• $f^{\prime }\in C_p(-\pi ,\pi )$¹

Then the Fourier series converges uniformly and absolutely to $f(x)$

Differentiation of Fourier series

Suppose that

• $f$ is continuous on $[-\pi ,\pi ]$
• $f(\pi )=f(-\pi )$
• $f^{\prime }\in C_p(-\pi ,\pi )$¹

Then the Fourier series is differentiable at $(-\pi <x<\pi )$ and if $f^{\prime \prime }(x)$ exist $f^{\prime }(x)=\sum _{n=1}^{\infty }(-n{A}_n\sin (nx)+n{B}_n\cos (nx))$

Integration of Fourier series

Suppose that

• $f$ is continuous on $(-\pi ,\pi )$
• $f(\pi )=f(-\pi )$
• $f^{\prime }\in C_p(-\pi ,\pi )$¹

Then the Fourier series representation for $f(x)$ on $(-\pi ,\pi )$ $F(x)={\int }_{-\pi }^{\pi }f(t)dt-\frac{A_0}{2}x$ continuous for $(-\pi <x<\pi )$ and
$F^{\prime }(x)=f(x)$ if $F$ is differentiable at $x$ We obtain that $F$ is continuous on $(-\pi <x<\pi )$ and $F^{\prime }\in C_p(-\pi ,\pi )$¹

Footnotes

1. Piecewise Continuous ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶ ↩⁷

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Piecewise Smooth

Definition

$C_p^{\prime }(a,b)$

function $f$ and $f^{\prime }$ are Piecewise Continuous on $(a,b)$

Facts

If $f\in C_p^{\prime }(a,b)$, then for each $x_0\in [a,b]$, $f_{+}^{\prime }(x_0)$ and $f_{-}^{\prime }(x_0)$ are exist.

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Absolute Convergence

Definition

A Series $\sum _{n=1}^{\infty }{a}_n$ converges absolutely if the series of absolute values $\sum _{n=1}^{\infty }|a_n|$ converges.

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Uniform Convergence

Definition

$\underset{n\to \infty }{\lim }{f}_n=f\text{ uniformly}$ Suppose $f:\mathcal{D}\to \mathbb{R}$, is a function, and $(f_n)$ is a Sequence of Functions whose term has the same domain as the function $f$ If $\forall ϵ>0,\exists N\in \mathbb{N},\forall x\in \mathcal{D},\forall n\ge N,|f_n(x)-f(x)|<ϵ$, then $(f_n)$ converges uniformly to $f$ on $\mathcal{D}$

Facts

Uniformly convergent Sequence of Functions $(f_n)$ implies Pointwise Convergence

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Weierstrass M-test

Definition

Let $(f_n)$ be a sequence of functions $f_n:\mathcal{D}\to \mathbb{R}$, and $(M_n)$ be a sequence of non-negative real numbers If $\exists M_n>0,\forall x\in \mathcal{D},\forall n\in \mathbb{N},(|f_n(x)|\le M_n\wedge \sum _{n=1}^{\infty }{M}_n<\infty )$, then $\sum _{n=1}^{\infty }{f}_n$ uniformly and absolutely converges on $\mathcal{D}$

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Superposition Principle

Definition

If $y_1,y_2$ are solutions of homogeneous linear ODE $y^{\prime \prime }+p(x)y^{\prime }+q(x)y=r(x)$ on $I=(\alpha ,\beta )$ then, any linear combination $c_1{y}_1+c_2{y}_2(c_1,c_2\in \mathbb{R})$ is also a solution (on $I$)

Facts

Suppose that each of infinitely many functions $u_1,u_2,\dots$ satisfies a linear homogeneous differential equation or boundary condition $L(u)=0$

Then, the infinite series $u={\sum }_{n=1}^{\infty }{c}_n{u}_n$ where $c_n\in \mathbb{R}\text{or}\mathbb{C}$ that satisfies following conditions satisfies $L(u)=0$

• the infinite series converges and required differentiability in $L(u)=0$
• the required boundary condition is satisfied

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Neumann problems

Definition

X'' (x) + \lambda X(x) = 0 \quad _{0<x<c},\quad X'(0)=0,\quad X'(c)=0

If eigenvalue $\lambda ={\lambda }_n={\frac{n^2{\pi }^2}{c^2}}_{(n=0,1,2,\dots )}$ then eigen function $X_n(x)=\cos {\frac{n\pi x}{c}}_{(n=0,1,2,\dots )}$ If $\lambda \ne {\lambda }_n{}_{(n=0,1,2,\dots )}$ then $X(x)=0$ is the only solution of problem

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Dirichlet problems

Definition

X'' (x) + \lambda X(x) = 0 \quad _{0<x<c},\quad X(0)=0,\quad X(c)=0

If eigenvalue $\lambda ={\lambda }_n={\frac{n^2{\pi }^2}{c^2}}_{(n=1,2,3,\dots )}$ then eigen function $X_n(x)=\sin {\frac{n\pi x}{c}}_{(n=0,1,2,\dots )}$ If $\lambda \ne {\lambda }_n{}_{(n=1,2,3,\dots )}$ then $X(x)=0$ is the only solution of problem

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Inner product

Definition

$⟨\cdot ,\cdot ⟩:V\times V\to K$ A function with the following properties

• $\forall \mathbf{u},\mathbf{v},\mathbf{w}\in V:⟨\mathbf{u}+\mathbf{v},\mathbf{w}⟩=⟨\mathbf{u},\mathbf{w}⟩+⟨\mathbf{v},\mathbf{w}⟩$^[Additivity]
• $\forall \mathbf{u},\mathbf{v}\in V,k\in K:⟨k\mathbf{u},\mathbf{v}⟩=k⟨\mathbf{u},\mathbf{v}⟩$^[Homogeneity]
• $\forall \mathbf{u},\mathbf{v}\in V:⟨\mathbf{u},\mathbf{v}⟩=\overline{⟨\mathbf{v},\mathbf{u}⟩}$^[Conjugate symmetry]
• $\forall 0\ne \mathbf{v}\in V:⟨\mathbf{v},\mathbf{v}⟩>0$^[Positive definiteness] where $K\in \{\mathbb{R},\mathbb{C}\}$ on a vector space $(V,+,\cdot ,K)$

Vector

Real

for $\mathbf{x},\mathbf{y}\in {\mathbb{R}}^n$ $⟨\mathbf{x},\mathbf{y}⟩=||\mathbf{x}|\cdot ||\mathbf{y}||\cos \theta ={\sum }_{i=1}^n{x}_i{y}_i$ where $||\cdot ||$ is a Norm

Complex

for $\mathbf{x},\mathbf{y}\in {\mathbb{C}}^n$ $⟨\mathbf{x},\mathbf{y}⟩=\mathbf{x}\overline{\mathbf{y}}={\sum }_{i=1}^n{x}_i{\bar{y}}_i$

Function

$⟨f,g⟩={\int }_a^{b}f(x)g(x)dx$ where $f,g\in C_p(a,b)$

Facts

$⟨f,g⟩=⟨g,f⟩$ $⟨f,g+h⟩=⟨f,g⟩+⟨f,h⟩$ $⟨cf,g⟩=c⟨f,g⟩,c\in \mathbb{R}$ $⟨f,f⟩\ge 0$ (by the definition of Norm)

${\mathbf{x}}^{⊺}\mathbf{y}=0\Rightarrow \mathbf{x}\perp \mathbf{y}$ ${\mathbf{x}}^{⊺}\mathbf{y}<0\Rightarrow \pi >\theta >\frac{1}{2}\pi$ ${\mathbf{x}}^{⊺}\mathbf{y}>0\Rightarrow 0<\theta <\frac{1}{2}\pi$

$\overline{\mathbf{x}}\mathbf{y}\notin \mathbb{R}\Rightarrow \overline{\overline{\mathbf{x}}\mathbf{y}}\ne \overline{\mathbf{y}}\mathbf{x}$

The inner product of Euclidean Space is called a dot product.

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Norm

Definition

$\Vert \cdot \Vert :V\to {\mathbb{R}}_0^{+}$ A real-valued function with the following properties

• Positive definiteness: $\forall \mathbf{v}\in V:\Vert \mathbf{v}\Vert =0\iff \mathbf{v}=0$
• Absolute Homogeneity: $\forall \mathbf{v}\in V,k\in K:\Vert k\mathbf{v}\Vert =|k|\Vert \mathbf{v}\Vert$
• Sub additivity or Triange inequality: $\forall \mathbf{v},\mathbf{w}\in V:\Vert \mathbf{v}+\mathbf{w}\Vert \le \Vert \mathbf{v}\Vert +\Vert \mathbf{w}\Vert$ where $K\in \{\mathbb{R},\mathbb{C}\}$ on a vector space $(V,+,\cdot ,K)$

Vector

Real

Let $\mathbf{x}\in {\mathbb{R}}^n$ be an $n$-dimensional vector, then the $L_p$ norm of the $\mathbf{x}$ is defined as $\Vert \mathbf{x}{\Vert }_p:={\left({\sum }_{i=1}^n|x_i{|}^p\right)}^{1/p}$

Complex

Let $\mathbf{x}\in {\mathbb{C}}^n$ be an $n$-dimensional complex vector, then the $L_1$ norm of the $\mathbf{x}$ is defined as $\Vert \mathbf{x}\Vert ={\mathbf{x}}^{†}\mathbf{x}=\sqrt{{\sum }_{i=1}^n{\bar{x}}_i{x}_i}$

Function

$\Vert f\Vert =\sqrt{⟨f,f⟩}=\sqrt{{\int }_a^{b}f(x{)}^{2}dx}$: norm of $f$

Facts

$\Vert \frac{f(x)}{\Vert f\Vert }\Vert =1$

a norm can be induced by a Inner product $d(v,w)=\Vert v-w\Vert$

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Orthogonality

Definition

Function

$f$ and $g$ are orthogonal if ${\int }_a^{b}f(x)g(x)dx=⟨f,g⟩$

Vector

$⟨\mathbf{v},\mathbf{w}⟩=0\Rightarrow \mathbf{v}\perp \mathbf{w}$

The two vectors are orthogonal if their Inner product is 0.

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Orthonormal Set

Definition

The set $\{{\varphi }_i|i\in \mathbb{N}\}\subset C_p(a,b)$^[Piecewise Continuous] satisfying $({\varphi }_m,{\varphi }_n)=\{\begin{array}{ll}0& m\ne n\\ 1& m=n\end{array}$

Facts

Closed Orthonormal set

If there is no function $f$ in $C_p(a,b)=X$ or in subspace of $X$ which is orthogonal to the $m$th orthonormal set $\{{\varphi }_m\}$ an orthonormal set $\{{\varphi }_n\}$ is closed in $C_p(a,b)=X$ or in subspace of $X$,

Bessel’s Inequality for Orthonormal set

Since MSE $E$ is non-negative, $||f|{|}^2-\sum _{n=1}^N{c}_n^2\ge 0\Rightarrow \sum _{n=1}^N{c}_n^2\le ||f|{|}^2$

Let $s_N=\sum _{n=1}^N{c}_n^2$ then a sequence of functions $\{s_N(x)\}{,}_{N\in \mathbb{N}}$ is bounded increasing sequence
So $\{s_N(x)\}$ converge to $s\in \mathbb{R}\Rightarrow \underset{n\to \infty }{\lim }{c}_n^2=0$

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Generalized Fourier series

Definition

Assume that $\{{\varphi }_n|n\in N\}\subset C_p(a,b)$ is an Orthonormal Set A function $f\in C_p(a,b)$ can be represented by a linear combination of $\{{\varphi }_n|n\in N\}$
$f(x)=\sum _{n=1}^{\infty }{c}_n{\varphi }_n(x)$, where $c_n={\int }_a^{b}f(x){\varphi }_n(x)dx$: generalized Fourier series

Facts

Best approximation in the mean

Let $f\in C_p(a,b)$ and $\{{\varphi }_n|n\in N\}$ an orthonormal set and ${\Phi }_N(x)=\sum _{n=1}^N{\gamma }_n{\varphi }_n(x)$
$E=||f-{\Phi }_N|{|}^2=||f|{|}^2+\sum _{n=1}^N({\gamma }_n-c_n{)}^2-\sum _{n=1}^N{c}_n^2$ where $c_n={\int }_a^{b}f(x){\varphi }_n(x)dx$
Therefore, the best approximation in the mean to $f(x)$ is $\sum _{n=1}^N{c}_n{\varphi }_n(x)$

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Complete Orthonormal Set

Definition

Suppose that $s_N=\sum _{n=1}^N{c}_n^2{\varphi }_n(x)$ and $Y\subset C_p(a,b)$.
If $\forall f\in Y$, $\underset{N\to \infty }{\lim }||f-s_N||=0$ is satisfied, we say $\{{\varphi }_n|n\in N\}$ is complete

Facts

Parseval’s Equation

Definition

$||f|{|}^2=\underset{N\to \infty }{\lim }{\sum }_{n=1}^N{c}_n^2=\underset{N\to \infty }{\lim }||{\sum }_{n=1}^N{c}_n{\varphi }_n|{|}^2$

Facts

Parseval’s equation is a sufficient and necessary condition for orthonormal set to be complete

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Sturm-Liouvile Boundary Value Problems

Definition

$[p(x)y^{\prime }{]}^{\prime }-q(x)y+\lambda r(x)y=0,0<x<1$ $a_{1}y(0)-a_2{y}^{\prime }(0)=0,b_{1}y(1)-b_2{y}^{\prime }(1)=0$

Facts

The differential operator $L[y]=-[p(x)y^{\prime }{]}^{\prime }+q(x)y\Rightarrow L[y]=\lambda r(x)y$
We assume that the functions $p,p^{\prime },q,r$ are continuous on $0\le x\le 1$
and $p(x)>0,r(x)>0$ on $[0,1]$

Lagrange identity: ${\int }_0^{1}L[u]vdx={\int }_0^{1}L[v]udx\iff (L[u],v)=(u,L[v])$

All of the eigenvalues and corresponding eigen functions are real

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