Numerical solution of Partial Differential Equations
- 1. Numerical solution of partial differential equations
- 2. Classification of Linear Second Order Partial Differential Equations When we model the realistic phenomena, we always have nonlinear partial differential equations that are not, in principle, solvable. For the solution of these equations numerically, we use finite difference methods and test these schemes for the standard PDEs. We develop finite difference schemes to parabolic, hyperbolic and elliptic equations.
- 3. It is well known that the differential equations that can be solved by explicit analytical formulae are very few. Therefore accurate numerical approximation schemes are essential for extracting quantitative information as well as qualitative understanding of the various behaviours of their solutions. The process includes: Replacing differential equations by suitable finite differences Applying boundary conditions Converting into system of equations Solutions of the algebraic equations Analysis of the solution numerically.
- 4. The general second order linear PDE has the following form: where the coefficients and the free term are in general functions of the independent variables but do not depend on the unknown function . The classification of second order equations depends on the form of the leading part of the equations consisting of the second order terms. The type of the above equation depends on the sign of the quantity which is called the discriminant of .
- 5. The classification of second order linear PDEs is given by the following. Elliptic equation if Parabolic equation if Hyperbolic equation if
- 6. Classify the following partial differential equations.
- 7. The simplest examples of the above equations are the following: Elliptic equation : (Two dimensional Laplace equation) Parabolic equation : (One dimensional heat equation) Hyperbolic equation : (One dimensional wave equation)
- 8. HEAT EQUATION The heat equation is an important partial differential equation which describes the distribution of heat (or variation in temperature) in a given region over time.
- 9. WAVE EQUATION The wave equation is one of the most important equations in mechanics. It describes not only the movement of strings and wires, but also the movement of fluid surfaces, e.g., water waves.
- 10. LAPLACE EQUATION The Laplace equations are used to describe the steady-state conduction heat transfer without any heat sources or sinks. Laplace equations can be used to determine the potential at any point between two surfaces when the potential of both surfaces is known. Now let us have a look at the different forms of Laplace equation examples in Physics. , the Laplace equation electrostatics defined for electric potential . If then, the Laplace equation in gravitational field. is the velocity of the steady flow.