The partial differential equation is a type of mathematical equation involving multivariable function and derivative. PDE is often used to model physical phenomena and complex systems, such as heat flow, waves, electromagnetism, fluid dynamics, and more. In this article, we're going to talk about the basic definition of PDE, the kind of PDE, and some general methods to solve it.
The partial differential equation is an equation involving the partial derivative of a function that depends on two or more independent variables. This equation is different from the common differential equation that only depends on one independent variable. For example, a simple PDE for one-dimensional heat flow is:
Here, u = u (x, t) u = u (x, t) u = u (x, t) is a function of xxx and time ttt, whereas negualphalez is a thermal conductive coefficient.
PDE occurs naturally when we model systems that involve change in space and time, or in systems with multiple dimensions.
PDE can be classified based on the nature of its equation:
PDE can also be classified based on the highest partial derivative that occurs in the equation. For example:
Solutions from PDE are often difficult to get analytically. Therefore, various methods, both analytic and numerical, have been developed to solve it. Here are some methods that are generally used to complete PDE:
Variable splitting methods are used to break the PDE into some simpler ODE. This technique is very useful when PDE has a fixed limit. For example, for a single-dimensional heat flow equation, we can assume that the solution is u (x, t) = x (t) t (x) t (x) t (x) t, where x (x) x (x) x (x) depends only on xxx (t) and t (t) t (t) just depends on tt. By separating variables like this, PDE can be reduced to two ODE.
Fourier and Laplace transformation methods are effective techniques to complete PDE linear with good boundary conditions. Fourier transformation, for example, is used to convert PDE into a space domain into a frequency domain, where equations become easier to solve. Once the equation is solved in a frequency domain, the solution can finally be returned to the original domain through transformation invers.
The method of elements up to is a numerical approach used to solve complex PDE, especially in the problems involving complex geometry and various border conditions. In this method, the physical domain of the problem is divided into small elements, and PDE is solved locally in each element. FEM is used a lot in engineering, especially in structure analysis and fluid dynamics.
The volume method up to another technique used in PDE numerical completion, especially in fluid dynamics and heat flow problems. This method works by dividing domains into discrete volume cells and completing integral of PDE in each of these small volumes. FVM is perfect for fluid flow problems because it guarantees eternal physical quantity, such as mass and energy, in each volume of control.
For PDE that depends on time, the renge- Kutta method is used as a numeric integration technique to solve equations in the time dimension. It works by solving PDE as an ordinary differential system for every step of time, which is very useful in time-based simulations, like waves and heat propagation.
The partial differential equation is used widely in many areas of science and engineering, including:
The partial differential equation is a very powerful mathematical tool to model dynamic phenomena in various areas of science. Although analytical solutions are often hard to come by, different methods such as variable separation, Fourier transformation, and numeric approaches like FEM allow us to solve PDE accurately and efficiently. PDE continues to play an important role in our understanding of the physical world and complex systems.
source: Strauss, W. A. Partials Diffential Equations: John Wiley & Sons.
