Ordinary Differential Equations on

Ordinary Differential Equations onhttps://bhavikdodda.github.io/amethyst/Math/ODE/Recent content in Ordinary Differential Equations onHugo -- gohugo.ioConvergence of functionshttps://bhavikdodda.github.io/amethyst/Math/ODE/Convergence-of-functions/Mon, 01 Jan 0001 00:00:00 +0000https://bhavikdodda.github.io/amethyst/Math/ODE/Convergence-of-functions/[[Uniform Convergence]] A sequence of functions {$f_n$} can be pointwise convergence to $f$ or uniformly continuous to $f$ If it’s uniformly continuous, it carries over some interesting properties: if $\forall n,f_{n}$ is continuous, then $f$ is also continuous $\lim_{n\rightarrow \infty}\int_{a}^{b}f_{n}dx=\int_{a}^{b}fdx$ series of functions # [!Definition] The infinite series $\sum\limits u_{n}(x)$ is said to be convergent uniformly if the sequence of partial sums {$f_{n}$} defined by $f_{n}(x)=\sum\limits_{i=1}^{n}u_{i}(x)$ is convergent uniformly on some interval for xTopology basicshttps://bhavikdodda.github.io/amethyst/Math/ODE/topology-stuff/Mon, 01 Jan 0001 00:00:00 +0000https://bhavikdodda.github.io/amethyst/Math/ODE/topology-stuff/Sets # [[2. Interval, Neighborhood, Interior]], [[3. Limit Point, Derived Set, Closure]], [[3.5 Compact, Dense, Perfect, Connected Sets]] Open Set, Closed Set, Bounded Set, Compact Set, Connected Set: Topology part-3 - YouTube Boundary Point # A point P is boundary point of domain D if every neighborhood of P contains a point in D and a point in D'. Connected Set # In the context of topology<link>https://bhavikdodda.github.io/amethyst/Math/ODE/Lipschitz-condition/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://bhavikdodda.github.io/amethyst/Math/ODE/Lipschitz-condition/</guid><description>Let f be defined on a [[Math/ODE/topology stuff#Domain|Domain]] or closed domain D of the XY plane. f is said to satisfy Lipschitz condition (wrt y) in D, if $$ \exists K(\forall (x,y_{1}),(x,y_{2})\in D(f(x,y_{1})-f(x,y_{2})\le K|y_{1}-y_{2}|)) $$ Sufficient condition # By [[Mean Value Theorems#First Mean Value Theorem]] of two variable functions, if $\frac{\partial f}{\partial y}$ is bounded by M, $$ \begin{align*} \exists E\in[y_{1},y_{2}]: f(x,y_{1})-f(x,y_{2})=(y_{1}-y_{2}) \frac{\partial f(x,E)}{\partial y}\ |f(x,y_{1})-f(x,y_{2})|<|y_{1}-y_{2}|M \end{align*} $$ Therefore, sufficient condition for Lipschitz condition is for $\frac{\partial f}{\partial y}$ to be bounded where lipscitz constant =</description></item><item><title>Differential Equations in 2 variableshttps://bhavikdodda.github.io/amethyst/Math/ODE/DE-in-two-var/Mon, 01 Jan 0001 00:00:00 +0000https://bhavikdodda.github.io/amethyst/Math/ODE/DE-in-two-var/$$ \frac{dy}{dx}=f(x,y) $$ where f is defined on a domain D and $(x_{0},y_{0})$ be an interior point of D. $\phi(x)$ is a solution of this DE IVP (Initial Value Problem) if it satisfies $[x,\phi(x)]\in D$ and thus $f[x,\phi(x)]$ is defined for $x\in [\alpha,\beta]$ containing $x_{0}$ $$\frac{d \phi(x)}{d x}=f(x,\phi(x))$$ $\phi(x_{0})=y_{0}$ Necessary and sufficient condition # $\phi(x)$ is a solution of the IVP over $[\alpha,\beta]$ if and only if it satisfies the integral equation: $$\phi(x)= y_{0}+\int_{x_{0}}^{x}f[t,\phi(t)]dt;\forall x\in [\alpha,\beta]$$<link>https://bhavikdodda.github.io/amethyst/Math/ODE/Existence-and-Uniqueness-Theorem/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://bhavikdodda.github.io/amethyst/Math/ODE/Existence-and-Uniqueness-Theorem/</guid><description>Sufficient condition for existence of solution # Consider a [[Math/ODE/DE in two var|DE in two var]]: $$ \frac{d y}{d x}=f(x,y) $$ Let f be continuous on D satisfies [[Math/ODE/Lipschitz condition|Lipschitz condition]] (wrt y) in D Let $(x_{0},y_{0})$ be an interior point in D and $a,b$ such that rectangle $|x-x_{0}|\le a,|y-y_{0}|\le b$ lies in D Let $M=\max(f(x,y))$ and $h=\min\left(a, \frac{b}{M}\right)$ If these are met, there exists a unique solution $\phi$ of the IVP on $|x-x_{0}|\le h$</description></item><item><title/><link>https://bhavikdodda.github.io/amethyst/Math/ODE/Gronwalls-Integral-Inequality/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://bhavikdodda.github.io/amethyst/Math/ODE/Gronwalls-Integral-Inequality/</guid><description>[!Theorem] Let g(x) and h(x) be continuous functions defined for $x>x_{0}$ such that $g(x),h(x)\ge 0$ Let k>0 be a constant such that $g(x)\le k+\int_{x_{0}}^{x}g(t)h(t)dt$ Then: $$g(x)\le ke^{\int_{x_{0}}^{x}h(t)dt}$$ Proof # $$ \begin{align*} \frac{g(x)h(x)}{k+\int_{x_{0}}^{x}g(t)h(t)dt}&\le h(x)\ \text{Integrating both sides}\ \left. \left(\log \left(k+\int_{x_{0}}^{x} g(t)h(t)dt\right)\right) \right|_{x_{0}}^{x}&\le \int_{x_{0}}^{x}h(t)dt\ \log\left(k+\int_{x_{0}}^{x}g(t)h(t)dt\right)-\log(k)&\le \int_{x_{0}}^{x}h(t)dt\ \ \left(k+\int_{x_{0}}^{x}g(t)h(t)dt\right)k^{-1}&\le e^{\int_{x_{0}}^{x}h(t)dt}\ \text{Using the given,}\ g(x)&\le k e^{\int_{x_{0}}^{x}h(t)dt} \end{align*} $$ Corollary # If $g(x)\ge0$ and k is a constant such that $k>0$, then if $g(x)\le k\int_{x_{0}}^{x}g(t)dt$, Then g(x)=0</description></item><item><title/><link>https://bhavikdodda.github.io/amethyst/Math/ODE/system-of-linear-odes/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://bhavikdodda.github.io/amethyst/Math/ODE/system-of-linear-odes/</guid><description>$$ \begin{align*} \frac{dx}{dt}=a_{1}(t)x+b_{1}(t)y+f_{1}(t)\ \frac{dy}{dt}=a_{2}(t)x+b_{2}(t)y+f_{2}(t) \end{align*} $$ Theorems # General sol # If homogeneous system has 2 linearly independent solutions $x_{1}(t),y_{1}(t)$ and $x_{2}(t),y_{2}(t)$ then general solution: $c_{1}x_{1}(t)+c_{2}x_{2}(t),c_{1}y_{1}(t)+c_{2}y_{2}(t)$ If non homogeneous solution has also a particular solution $x_{p}(t),y_{p}(t)$, then general solution: $c_{1}x_{1}(t)+c_{2}x_{2}(t)+x_{p}(t),c_{1}y_{1}(t)+c_{2}y_{2}(t)+y_{p}(t)$ derivative of wronskian # $$ \begin{align*} \frac{d}{dt}W[t]&= \ \ &= [a_{1}+b_{2}]W \end{align*} $$ It is either 0 or “nowhere vanishing”. Solving # constant coeff # We assume the solution of form $x(t)=Ae^{mt};y(t)=Be^{mt}$ and get an auxilary equation in terms of m</description></item><item><title/><link>https://bhavikdodda.github.io/amethyst/Math/ODE/Non-Linear-Theory/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://bhavikdodda.github.io/amethyst/Math/ODE/Non-Linear-Theory/</guid><description>General # $$ \begin{align*} \frac{dx}{dt}=F(x,y)\ \frac{dy}{dt}=G(x,y) \end{align*} $$ Solution to this is a directed curve. The critical points can be found by solving $F(x,y)=0,G(x,y)=0$ Types of critical points Node Stable Unstable Saddle point Center Spiral Critical Points and Stability for linear system # Depends on auxilary roots of [[Math/ODE/system of linear odes|system of linear odes]] $$ \begin{align*} \frac{dx}{dt}&= a_{1}x+b_{1}y\ \frac{dy}{dt}&= a_{2}x+b_{2}y\ \ \text{Auxilary quadratic equation}\ m^{2}-(a_{1}+b_{2})m+(a_{1}b_{2}-a_{2}b_{1})=0\ x=Ae^{mt};y=Be^{mt} \end{align*} $$</description></item><item><title/><link>https://bhavikdodda.github.io/amethyst/Math/ODE/Boundary-Value-Problems/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://bhavikdodda.github.io/amethyst/Math/ODE/Boundary-Value-Problems/</guid><description>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</description></item><item><title/><link>https://bhavikdodda.github.io/amethyst/Math/ODE/Greens-Function/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://bhavikdodda.github.io/amethyst/Math/ODE/Greens-Function/</guid><description>Green’s Function # L(y)=f(x) A function G(x,t) defined for $a\le x\le t$, $t\le x \le b$ is called green’s function of L(y) if it satisfies: G(x,t) has derivatives upto order n. $$ \frac{\partial ^{k}G}{\partial x^{k}}|_{t^{-}}^{t^{+}}=0,k=0,1,2,…n-2 $$ $$\frac{\partial ^{k}G}{\partial x^{k}}|_{t^{-}}^{t^{+}}= 1/c_0(t) ,k=n-1$$ G(x,t) considered as a function of x, is a solution of L(y)=0. L(G)=0 on each of the intervals. It satisfies linear homogeneous boundary conditions Theorem of unique green function # [!</description></item><item><title/><link>https://bhavikdodda.github.io/amethyst/Math/ODE/Non-Homogeneous-Linear-Systems/</link><pubDate>Mon, 01 Jan 0001 00:00:00 +0000</pubDate><guid>https://bhavikdodda.github.io/amethyst/Math/ODE/Non-Homogeneous-Linear-Systems/</guid><description>$$ \begin{align*} \text{DE: } \frac{dX}{dt}&= A(t)X+F(x)\ \text{A solution: }\phi_{0}(t)&= \phi(t)\int_{t_{0}}^{t}\phi^{-1}(u)F(u)du\ \text{where fundamental matrix of corresponding homogeneous: }\phi(t)\\ \text{General solution satisfying BVP }X(t_{0})=x_{0}\ X(t)&= \phi(t)\phi^{-1}(t_{0})x_{0}+ \phi_{0}(t)\ \end{align*} $$</description></item><item><title>Table of Contentshttps://bhavikdodda.github.io/amethyst/Math/ODE/ODE/Mon, 01 Jan 0001 00:00:00 +0000https://bhavikdodda.github.io/amethyst/Math/ODE/ODE/ToC # [[Math/ODE/Convergence of functions|Convergence of functions]] [[Math/ODE/topology stuff|topology stuff]] [[Math/ODE/Lipschitz condition|Lipschitz condition]] [[Math/ODE/DE in two var|DE in two var]] [[Math/ODE/Existence and Uniqueness Theorem|Existence and Uniqueness Theorem]] [[Math/ODE/Gronwalls Integral Inequality|Gronwalls Integral Inequality]] [[Math/ODE/system of linear odes|system of linear odes]] [[Math/ODE/Non Linear Theory|Non Linear Theory]] [[Math/ODE/Boundary Value Problems|Boundary Value Problems]] [[Math/ODE/Greens Function|Greens Function]] [[Math/ODE/Non Homogeneous Linear Systems|Non Homogeneous Linear Systems]] Sources # Class notes