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Boundary Value Problems

Linear differential equation:

$$ L(y)=c_{0}(x) \frac{d^{n}y}{dx^{n}}+c_{1}(x) \frac{d^{n-1}y}{dx^{n-1}}+…+c_{n}y=f(x) $$

where $c_{i}$ are real or complex valued continuous functions on an interval I=$[a,b]$

Periodic boundary conditions

$$ y(a)=y(b),y’(a)=y’(b),…,y^{n-1}(a)=y^{n-1}(b) $$

Linear forms stated at ‘a’ and ‘b’

$$ v_{k}(y)= \left(\alpha_{k}^{0}y(a)+ \alpha_{k}^{1} \frac{dy}{dx}(a)+…+ \alpha_{k}^{n-1} \frac{d^{n-1}}{dx^{n-1}}(a)\right)+\left(\beta_{k}^{0}y(b)+ \beta_{k}^{1} \frac{dy}{dx}(b)+…+ \beta_{k}^{n-1} \frac{d^{n-1}}{dx^{n-1}}(b)\right) $$

Homogeneous linear forms if $\forall k(v_{k}(y)=0)$

Classification #

Linear homogeneous BVP #

L(y)=0 satisfying homogeneous linear forms $v_{k}(y)=0$

Linear non-homogenous BVP #

If the linear forms are linearly independent, then finding a solution to $L(y)=f(x)$ is called a linear non-homogeneous BVP

Regular BVP #

A linear BVP (homogeneous or non-homogeneous) is called a regular BVP if

  1. a,b are finite
  2. $c_{0}(x)\ne0$ on $[a,b]$

Singular BVP #

A linear BVP is singular if it’s not regular.

Vital BVPs #

$$ \frac{d^{2}y}{dx^{2}}=f(x,y,\frac{dy}{dx}) $$

First BVP #

Solving the Vital BVP with boundary conditions

$$ y(a)=\alpha,y(b)=\beta $$

Equation has preassigned values at a,b

Second BVP #

Solving the Vital BVP with boundary conditions

$$ \begin{align*} y(a)=\alpha_{1},y’(b)=\alpha_{2}\\ \text{OR}\\ y’(a)=\beta_{1},y(b)=\beta_{2} \end{align*} $$

Equation with preassigned value at one point and a prescribed slope at the other

Third BVP #

Solving the Vital BVP with boundary conditions

$$ y’(a)=\alpha,y’(b)=\beta $$

Equation that has prescribed slopes at a and b.

Strum Liouville Problems SLP #

$$ \frac{d}{dx}(r(x) \frac{dy}{dx})+(q(x)+\lambda p(x))y=0 $$

r,q,p,r’ are real-valued continuous functoins defined on interval I.

$\lambda$ is a real parameter.

Boundary Conditions #

$$ \begin{align*} k_{1}y(a)+k_{2} \frac{dy}{dx}(a)=0\\ \text{OR}\\ k_{3}y(b)+k_{4} \frac{dy}{dx}(b)=0\\ \\ \text{Periodic boundary conditions}\\ y(a)=y(b),y’(a)=y’(b),r(a)=r(b) \end{align*} $$

Eivenvalue and Eigenfunction #

y(x)=0 is a trivial solution for Strum Lioville.

$\lambda$ for which it has a non-trivial solution is called eigenvalue and non-trivial solution associated with it is called eigenfunction

Representation of other DE in SLP #

Legendre equation #

$$ \begin{align*} (1-x^{2}) \frac{d^{2}y}{dx^{2}}-2x \frac{dy}{dx}+ n(n+1)y=0\\ \frac{d}{dx}((1-x^{2}) \frac{dy}{dx})+ \lambda y=0\\ \lambda=n(n+1) \end{align*} $$

Bessel’s equation #

Example #

$$ \begin{align*} \frac{d^{2}y}{dx^{2}}+ \lambda y=0\\ y(0)=0,y(\pi)=0\\ \\ \text{Eigenvalues: }\lambda=n^{2} \text{ where }n\in \mathbb{N}\\ \text{Eigenfunctions: }a_{n}\sin (\sqrt \lambda x) \end{align*} $$

Theorem #

The differential equation:

Strum Liouville Problems SLP

With the conditions:

$$ \begin{align*} k_{1}y(a)+k_{2}y’(a)=0\\ k_{3}y(b)+k_{4}y’(b)=0 \end{align*} $$

Conclusion:

  • There exists an infinite number of eigenvalues $\lambda_{n}$ and it can be arranged in monotonic increasing sequence.
  • For each $\lambda_{n}$ there exists one-parameter family of eigenfunctions $\phi_{n}$ and any 2 eigenfunctions corresponding to the same eigenvalue are nonzero constant multiples of each other.
  • Each eigenfunction $\phi_{n}$ has exactly (n-1) zeroes in the interval (a,b)

Characteristic Value and Characteristic function #

Characteristic value, function = eigen value, function

Proof:

Consider $\lambda= \alpha+ i \beta$ and corresponding eigenfunction $y(x)+u(x)+iv(x)$

Substituting in the SLP and then equating the real and imaginary parts of the DE to 0, we obtain that $\beta=0$. Therefore, $\lambda$ is real.

Existence #

Eigenvalues $\lambda$ are real valued

Orthogonality of eigenfunctions #

Definition #

$\phi_{1},\phi_{2}$ are orthogonal with respect to weight function $p(x)>0$ if

$$ \int_{a}^{b} p(x)\phi_{1}(x)\phi_{2}(x)dx=0 $$

Norm of $\phi_{1}$

$$ ||\phi_{1}||=\sqrt{\int p(x) \phi_{1}^{2}(x)dx} $$

Orthonormal: if the functions are orthogonal and the norm of each one = 1

Theorem #

Eigen functions of SLP (Strum Liouville Problems SLP) $\phi_{m},\phi_{n}$ are orthogonal with respect to p(x)

Boundary conditions that can be dropped

  1. If r(a)=0
    1. $k_{1}y(a)+k_{2}y’(a)=0$ can be dropped
  2. If r(b)=0
    1. $k_{3}y(b)+k_{4}y’(b)=0$ can be dropped
  3. If r(a)=r(b)
    1. Boundary conditions can be replaced by periodic $y(a)=y(b),y’(a)=y’(b)$
    2. called Periodic Strum-Liouville problem

Proof #

Let $\phi_{m},\phi_{n}$ be the eigenfunctions satisfying SLP for eigenvalues $\lambda_{m},\lambda_{n}$

$$ \begin{align*} \frac{d}{dx}(r(x) \frac{d \phi_{m}}{dx})+(q(x)+\lambda_{m}p(x))\phi_{m}(x)=0\\ \frac{d}{dx}(r(x) \frac{d \phi_{n}}{dx})+(q(x)+ \lambda_{n}p(x))\phi_{n}(x)=0\\ \\ (\lambda_{m}-\lambda_{n})\int_{a}^{b}p(x)\phi_{m}(x)\phi_{n}(x)dx&= [r(x)(\frac{d \phi_{n}}{dx}\phi_{m}(x)- \frac{d \phi_{m}}{dx}\phi_{n}(x))]_{a}^{b} \end{align*} $$

If r(a) and r(b) = 0 #

RHS automatically becomes 0.

LHS=0, therefore the eigenfunctions are orthogonal

If r(a)=0 only #

Utilize the following to make the RHS=0

$$ \begin{align*} k_{3}\phi_{n}(b)+k_{4}\phi_{n}’(b)=0\\ k_{3}\phi_{m}(b)+k_{4}\phi_{m}’(b)=0 \end{align*} $$

If r(b)=0 only #

Utilize the following to make the RHS=0

$$ \begin{align*} k_{1}\phi_{n}(a)+k_{2}\phi_{n}’(a)=0\\ k_{1}\phi_{m}(a)+k_{2}\phi_{m}’(a)=0 \end{align*} $$

If both are non zero #

We use both the boundary conditions and proceed.

Examples #

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