# aanshik avakal sameekaran

ganit mein aanshik avakal sameekaran vo avakal sameekaranein hoti hain jinmein bahuchar falan aur unke aanshik avakal hote hain. (yeh saadhaaran avakal sameekaranon se bhinn hai jinmein ek hi char aur uske avakalon mein banta hua hota hai. aanshik avakal sameekaranon ka upayog un samasyaaon ko hal karne mein prayukt kiya jaata hai jo vibhinn svatantr charon ki falan hoti hain evam jinhein saadhaaranataya hal kar sakte hain athva hal karne ke liye abhiklitr program banaaya ja sake.

aanshik avakal sameekarano ka upayog vibhinn drushtigt ghatnaaon yatha dhvani, ooshma, sthirvaidyutiki, vidyut-gatiki, drav ka pravaah, pratyaasthata ya pramaatra yaantriki ko samajhne mein kiya ja sakta hai. ye pruthak prateet hone waali prakriyaaon ko aanshik avakal sameekaranon ke roop mein sootrit kiya ja sakta hai.

## udaaharan

nimnalikhit sameekaran, aanshik avakal sameekaran ka ek udaaharan hai-

${\frac {\partial ^{2}u}{\partial x\partial y}}+{\frac {\partial u}{\partial x}}=-{\frac {y}{x}}$ ,

jiska saamaanya hal (general saloosan) nimnalikhit hai-

$u(x,y)=F(y)+e^{-y}G(x)-\ln(x)\left(y-1\right)$ .

jahaaain $F$ aur $G$ yaadruchhik (aarbitreri) falan hain.

### laaplaas ka sameekaran

${\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}+{\partial ^{2}u \over \partial z^{2}}=0$ ### tani hui dori ka kampan

${\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}+{\partial ^{2}u \over \partial z^{2}}={1 \over c^{2}}{\partial ^{2}u \over \partial t^{2}}$ ### furre (Fourier) ka sameekaran

${\partial ^{2}u \over \partial x^{2}}+{\partial ^{2}u \over \partial y^{2}}+{\partial ^{2}u \over \partial z^{2}}={1 \over \alpha }{\partial u \over \partial t}$ ${\frac {\partial u}{\partial t}}+a{\frac {\partial u}{\partial x}}=0$ ### laingamoor (Langmuir ) ka sameekaran

$\Delta \psi \ ={1 \over c^{2}}.{\partial ^{2}\psi \over \partial t^{2}}-{\rho \over \epsilon }$ ### stoks (Stokes) ka sameekaran

$\eta \Delta {\vec {v}}={\overrightarrow {\mathrm {grad} }}\,p-\rho {\vec {f}}$ ,

$i\hbar {\partial \psi \over \partial t}\ =\left[-{\frac {\hbar ^{2}}{2m}}\Delta +V\right]\psi$ ### clean aur gaurdan (Klein-Gordon )ka sameekaran

$-\hbar ^{2}{\partial ^{2}\psi \over \partial t^{2}}\ =-\hbar ^{2}c^{2}\Delta \psi +m^{2}c^{4}\psi$ ## vargeekaran

do order vaale aanshik avakal sameekaranon ko paravalayi (parabolic), ativlayi (hyperbolic) aur deerghavrutteeya (elliptic) mein vibhkt kiya jaata hai.

$u_{xy}=u_{yx}$ ko maanate hue, maana do svatantr charon mein, do-order wala, saamaanya PDE nimnalikhit hai-

$Au_{xx}+2Bu_{xy}+Cu_{yy}+\cdots {\mbox{(lower order terms)}}=0,$ jahaaain A, B, C aadi gunaank x aur y par nirbhar ho sakte haim. yadi xy-plane ke kisi kshetr mein $A^{2}+B^{2}+C^{2}>0$ ho, to us kshetr mein PDE dviteeya-order wala hai. yeh roop shaankav (conic section) ke sameekaran jaisa hai:

$Ax^{2}+2Bxy+Cy^{2}+\cdots =0.$ doosare shabdon mein, , ∂x ke sthaan par X rakhane par, (aur isi prakaar anya charon ke liye bhi karne par) niyat gunaank wala PDE usi degree ke ek bahupad mein parivrtit ho jaata hai.

jis prakaar diskriminent $B^{2}-4AC$ ke aadhaar par konik sekshans ko parabolic, hyperbolic, aur elliptic mein baaainta jaata hai, usi tarah dviteeya-order vaale PDE ko bhi vargeekrut kiya ja sakta hai. kintu PDE ke case mein diskriminent $B^{2}-AC,$ liya jaata hai.

1. $B^{2}-AC<0$ : solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth. The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where x < 0.
2. $B^{2}-AC=0$ : equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where x = 0.
3. $B^{2}-AC>0$ : hyperbolic equations retain any discontinuities of functions or derivatives in the initial data. An example is the wave equation. The motion of a fluid at supersonic speeds can be approximated with hyperbolic PDEs, and the Euler–Tricomi equation is hyperbolic where x > 0.

If there are n independent variables x1, x2 , ..., xn, a general linear partial differential equation of second order has the form

$Lu=\sum _{i=1}^{n}\sum _{j=1}^{n}a_{i,j}{\frac {\partial ^{2}u}{\partial x_{i}\partial x_{j}}}\quad {\text{ plus lower-order terms}}=0.$ The classification depends upon the signature of the eigenvalues of the coefficient matrix ai,j..

1. Elliptic: The eigenvalues are all positive or all negative.
2. Parabolic : The eigenvalues are all positive or all negative, save one that is zero.
3. Hyperbolic: There is only one negative eigenvalue and all the rest are positive, or there is only one positive eigenvalue and all the rest are negative.
4. Ultrahyperbolic: There is more than one positive eigenvalue and more than one negative eigenvalue, and there are no zero eigenvalues. There is only limited theory for ultrahyperbolic equations (Courant and Hilbert, 1962).