"आंशिक अवकल समीकरण": अवतरणों में अंतर
Content deleted Content added
अनुनाद सिंह (वार्ता | योगदान) |
अनुनाद सिंह (वार्ता | योगदान) |
||
पंक्ति 46:
:<math>-\hbar^2 {\part^2 \psi \over \part t^2} \ = - \hbar^2 c^2 \Delta \psi + m^2c^4 \psi </math>
==वर्गीकरण==
दो ऑर्डर वाले आंशिक अवकल समीकरणों को परवलयी (parabolic), अतिवलयी (hyperbolic) और दीर्घवृत्तीय (elliptic) में विभक्त किया जाता है।
<math>u_{xy}=u_{yx}</math> को मानते हुए, माना दो स्वतन्त्र चरों में, दो-ऑर्डर वाला, सामान्य PDE निम्नलिखित है-
: <math>Au_{xx} + 2Bu_{xy} + Cu_{yy} + \cdots \mbox{(lower order terms)} = 0,</math>
जहाँ ''A'', ''B'', ''C'' आदि गुणांक ''x'' और ''y'' पर निर्भर हो सकते हैम। यदि xy-प्लेन के किसी क्षेत्र में <math>A^2 +B^2 + C^2 > 0</math> हो, तो उस क्षेत्र में PDE द्वितीय-ऑर्डर वाला है। यह रूप [[शांकव]] (conic section) के समीकरण जैसा है:
: <math>Ax^2 + 2Bxy + Cy^2 + \cdots = 0.</math>
दूसरे शब्दों में, , ∂<sub>''x''</sub> के स्थान पर ''X'' रखने पर, (और इसी प्रकार अन्य चरों के लिये भी करने पर) नियत गुणांक वाला PDE उसी डिग्री के एक [[बहुपद]] में परिवर्तित हो जाता है।
जिस प्रकार डिस्क्रिमिनेन्ट <math>B^2 - 4AC</math> के आधार पर कोनिक सेक्शन्स को parabolic, hyperbolic, और elliptic में बाँटा जाता है, उसी तरह द्वितीय-ऑर्डर वाले PDE को भी वर्गीकृत किया जा सकता है। किन्तु PDE के केस में डिस्क्रिमिनेन्ट <math>B^2 - AC,</math> लिया जाता है।
# <math>B^2 - AC < 0</math>: solutions of [[elliptic partial differential equation|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.
# <math>B^2 - AC = 0</math>: equations that are [[parabolic partial differential equation|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.
# <math>B^2 - AC > 0 </math>: [[hyperbolic partial differential equation|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 ''x''<sub>1</sub>, ''x''<sub>2 </sub>, ..., ''x''<sub>''n''</sub>, a general linear partial differential equation of second order has the form
: <math>L u =\sum_{i=1}^n\sum_{j=1}^n a_{i,j} \frac{\part^2 u}{\partial x_i \partial x_j} \quad \text{ plus lower-order terms} =0.</math>
The classification depends upon the signature of the eigenvalues of the coefficient matrix ''a<sub>i,j</sub>''..
# Elliptic: The eigenvalues are all positive or all negative.
# Parabolic : The eigenvalues are all positive or all negative, save one that is zero.
# 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.
# 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).
== सन्दर्भ ==
| |||