Парцијална диференцијална једначина

Визуализација решења дводимензионалне једначине топлоте са температуром која је представљена трећом димензијом

Парцијална диференцијална једначина је диференцијална једначина која садржи претходно непознате функције са више променљивих и њихове парцијалне изводе. Користе се за формулисање проблема који укључују функције више променљивих, а решавају се ручно или се користе за креирање компјутерских модела. Посебан случај су обичне диференцијалне једначине које се баве функцијама једне променљиве и њиховим изводима.

Парцијалне диференцијалне једначине могу се користити за опис широког спектра феномена као што су звук, дифузија, топлота, електростатика, електродинамика, динамика флуида, еластичност или квантна механика. Баш као што обичне диференцијалне једначине често моделирају једнодимензионалне динамичке системе, парцијалне диференцијалне једначине често моделирају вишедимензионалне системе. Парцијалне диференцијалне једначине проналазе своју генерализацију у стохастичким парцијалним диференцијалним једначинама.

• Линеарна хомогена парцијална једначина је облика:

$P_{1}(x_{1},...,x_{n}){\frac {\partial u}{\partial x_{1}}}+...+P_{n}(x_{1},...,x_{n}){\frac {\partial u}{\partial x_{n}}}=0$.

• Квазилинеарна парцијална једначина је облика:

$P_{1}(x_{1},...,x_{n},u){\frac {\partial u}{\partial x_{1}}}+...+P_{n}(x_{1},...,x_{n},u){\frac {\partial u}{\partial x_{n}}}=P_{n+1}(x_{1},...,x_{n},u)$.

Увод

Парцијалне диференцијалне једначине (ПДЕ) су једначина које involve rates of change with respect to continuous variables. For example, the position of a rigid body is specified by six parameters, but the configuration of a fluid is given by the continuous distribution of several parameters, such as the temperature, pressure, and so forth. The dynamics for the rigid body take place in a finite-dimensional configuration space; the dynamics for the fluid occur in an infinite-dimensional configuration space. This distinction usually makes PDEs much harder to solve than ordinary differential equations (ODEs), but here again, there will be simple solutions for linear problems. Classic domains where PDEs are used include acoustics, fluid dynamics, electrodynamics, and heat transfer.

Парцијална диференцијална једначина за функцију u(x₁,… x_n) is an equation of the form

$f\left(x_{1},\ldots x_{n};u,{\frac {\partial u}{\partial x_{1}}},\ldots {\frac {\partial u}{\partial x_{n}}};{\frac {\partial ^{2}u}{\partial x_{1}\partial x_{1}}},\ldots {\frac {\partial ^{2}u}{\partial x_{1}\partial x_{n}}};\ldots \right)=0.$

If f is a linear function of u and its derivatives, then the PDE is called linear. Common examples of linear PDEs include the heat equation, the wave equation, Laplace's equation, Helmholtz equation, Klein–Gordon equation, and Poisson's equation.

A relatively simple PDE is

${\frac {\partial u}{\partial x}}(x,y)=0.$

This relation implies that the function u(x,y) is independent of x. However, the equation gives no information on the function's dependence on the variable y. Hence the general solution of this equation is

$u(x,y)=f(y),$

where f is an arbitrary function of y. The analogous ordinary differential equation is

${\frac {\mathrm {d} u}{\mathrm {d} x}}(x)=0,$

which has the solution

$u(x)=c,$

where c is any constant value. These two examples illustrate that general solutions of ordinary differential equations (ODEs) involve arbitrary constants, but solutions of PDEs involve arbitrary functions. A solution of a PDE is generally not unique; additional conditions must generally be specified on the boundary of the region where the solution is defined. For instance, in the simple example above, the function f(y) can be determined if u is specified on the line x = 0.

Постојање и јединственост

Док питање постојања и јединствености решења обичних диференцијалних једначина има веома задовољавајуће показатеље уз примену Picard–Lindelöf theorem, that is far from the case for partial differential equations. The Cauchy–Kowalevski theorem states that the Cauchy problem for any partial differential equation whose coefficients are analytic in the unknown function and its derivatives, has a locally unique analytic solution. Although this result might appear to settle the existence and uniqueness of solutions, there are examples of linear partial differential equations whose coefficients have derivatives of all orders (which are nevertheless not analytic) but which have no solutions at all: see Lewy (1957). Even if the solution of a partial differential equation exists and is unique, it may nevertheless have undesirable properties. The mathematical study of these questions is usually in the more powerful context of weak solutions.

An example of pathological behavior is the sequence (depending upon n) of Cauchy problems for the Laplace equation

${\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}=0,$

with boundary conditions

${\begin{aligned}u(x,0)&=0,\\{\frac {\partial u}{\partial y}}(x,0)&={\frac {\sin nx}{n}},\end{aligned}}$

where n is an integer. The derivative of u with respect to y approaches zero uniformly in x as n increases, but the solution is

$u(x,y)={\frac {\sinh ny\sin nx}{n^{2}}}.$

This solution approaches infinity if nx is not an integer multiple of π for any non-zero value of y. The Cauchy problem for the Laplace equation is called ill-posed or not well-posed, since the solution does not continuously depend on the data of the problem. Such ill-posed problems are not usually satisfactory for physical applications.

Нотација

У парцијалним диференцијалним једначинама је уобичајено да се парцијални деривати означе користећи индексе.

$u_{x}={\frac {\partial u}{\partial x}}$

$u_{xx}={\frac {\partial ^{2}u}{\partial x^{2}}}$

$u_{xy}={\frac {\partial ^{2}u}{\partial y\,\partial x}}={\frac {\partial }{\partial y}}\left({\frac {\partial u}{\partial x}}\right).$

У физици се, del или набла (∇) често користе за означавање просторних извода, а u̇, ü за временске изводе. На пример, таласна једначина (доле описана) се може написати као

${\ddot {u}}=c^{2}\nabla ^{2}u$

или

${\ddot {u}}=c^{2}\Delta u$

где је Δ Лапласов оператор.

Види још

• Диференцијална једначина

Литература

• Adomian, G. (1994). Solving Frontier problems of Physics: The decomposition method. Kluwer Academic Publishers. 
• Courant, R. & Hilbert, D. (1962), Methods of Mathematical Physics, II, New York: Wiley-Interscience .
• Evans, L. C. (1998), Partial Differential Equations, Providence: American Mathematical Society, ISBN 0-8218-0772-2 .
• Holubová, Pavel Drábek ; Gabriela (2007). Elements of partial differential equations ([Online-Ausg.]. изд.). Berlin: de Gruyter. ISBN 9783110191240. 
• Ibragimov, Nail H (1993), CRC Handbook of Lie Group Analysis of Differential Equations Vol. 1-3, Providence: CRC-Press, ISBN 0-8493-4488-3 .
• John, F. (1982), Partial Differential Equations (4th изд.), New York: Springer-Verlag, ISBN 0-387-90609-6 .
• Jost, J. (2002), Partial Differential Equations, New York: Springer-Verlag, ISBN 0-387-95428-7 .
• Lewy, Hans (1957), „An example of a smooth linear partial differential equation without solution”, Annals of Mathematics, Second Series, 66 (1): 155—158, doi:10.2307/1970121 .
• Liao, S.J. (2003), Beyond Perturbation: Introduction to the Homotopy Analysis Method, Boca Raton: Chapman & Hall/ CRC Press, ISBN 1-58488-407-X 
• Olver, P.J. (1995), Equivalence, Invariants and Symmetry, Cambridge Press .
• Petrovskii, I. G. (1967), Partial Differential Equations, Philadelphia: W. B. Saunders Co. .
• Pinchover, Y. & Rubinstein, J. (2005), An Introduction to Partial Differential Equations, New York: Cambridge University Press, ISBN 0-521-84886-5 .
• Polyanin, A. D. (2002), Handbook of Linear Partial Differential Equations for Engineers and Scientists, Boca Raton: Chapman & Hall/CRC Press, ISBN 1-58488-299-9 .
• Polyanin, A. D. & Zaitsev, V. F. (2004), Handbook of Nonlinear Partial Differential Equations, Boca Raton: Chapman & Hall/CRC Press, ISBN 1-58488-355-3 .
• Polyanin, A. D.; Zaitsev, V. F. & Moussiaux, A. (2002), Handbook of First Order Partial Differential Equations, London: Taylor & Francis, ISBN 0-415-27267-X .
• Roubíček, T. (2013), Nonlinear Partial Differential Equations with Applications (2nd изд.), Basel, Boston, Berlin: Birkhäuser, ISBN 978-3-0348-0512-4, MR 3014456, doi:10.1007/978-3-0348-0513-1 
• Solin, P. (2005), Partial Differential Equations and the Finite Element Method, Hoboken, NJ: J. Wiley & Sons, ISBN 0-471-72070-4 .
• Solin, P.; Segeth, K. & Dolezel, I. (2003), Higher-Order Finite Element Methods, Boca Raton: Chapman & Hall/CRC Press, ISBN 1-58488-438-X .
• Stephani, H. (1989), Differential Equations: Their Solution Using Symmetries. Edited by M. MacCallum, Cambridge University Press .
• Wazwaz, Abdul-Majid (2009). Partial Differential Equations and Solitary Waves Theory. Higher Education Press. ISBN 978-3-642-00251-9. 
• Wazwaz, Abdul-Majid (2002). Partial Differential Equations Methods and Applications. A.A. Balkema. ISBN 90-5809-369-7. 
• Zwillinger, D. (1997), Handbook of Differential Equations (3rd изд.), Boston: Academic Press, ISBN 0-12-784395-7 .
• Gershenfeld, N. (1999), The Nature of Mathematical Modeling (1st изд.), New York: Cambridge University Press, New York, NY, USA, ISBN 0-521-57095-6 .
• Krasil'shchik, I.S. & Vinogradov, A.M., Eds. (1999), Symmetries and Conserwation Laws for Differential Equations of Mathematical Physics, American Mathematical Society, Providence, Rhode Island, USA, ISBN 0-8218-0958-X .
• Krasil'shchik, I.S.; Lychagin, V.V. & Vinogradov, A.M. (1986), Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach Science Publishers, New York, London, Paris, Montreux, Tokyo, ISBN 2-88124-051-8 .
• Vinogradov, A.M. (2001), Cohomological Analysis of Partial Differential Equations and Secondary Calculus, American Mathematical Society, Providence, Rhode Island, USA, ISBN 0-8218-2922-X .
• Cajori, Florian (1928). „The Early History of Partial Differential Equations and of Partial Differentiation and Integration” (PDF). The American Mathematical Monthly. 35 (9): 459—467. doi:10.2307/2298771. 

Спољашње везе

• Hazewinkel Michiel, ур. (2001). „Differential equation, partial”. Encyclopaedia of Mathematics. Springer. ISBN 978-1556080104. 
• Partial Differential Equations: Exact Solutions at EqWorld: The World of Mathematical Equations.
• Partial Differential Equations: Index at EqWorld: The World of Mathematical Equations.
• Partial Differential Equations: Methods at EqWorld: The World of Mathematical Equations.
• Example problems with solutions at exampleproblems.com
• Partial Differential Equations at mathworld.wolfram.com
• Partial Differential Equations with Mathematica
• Partial Differential Equations in Cleve Moler: Numerical Computing with MATLAB
• Partial Differential Equations at nag.com
• Dispersive PDE Wiki
• NEQwiki, the nonlinear equations encyclopedia
• Partial differential equation | Scholarpedia