Chapter 7 Partial Differential Equation (PDE)
The equations which contain partial differential coefficients, the independent variables and dependent variable are known as partial differential equations. In other words, a partial differential equation (PDE) relates a function of multiple variables to its partial derivatives. It involves the derivatives of the unknown function with respect to two or more independent variables. The order of partial differential equaion is the same as the order of highest derivative occuring in the equation. If z is taken as dependent variable and x, y, and t as independent variables then the equation
\begin{equation*}
y\frac{\partial z}{\partial x}+x\frac{\partial z}{\partial y}+\frac{\partial z}{\partial t} =0
\end{equation*}
is the first order PDE in three variables, the equation
\begin{equation*}
\left(\frac{\partial z}{\partial x}\right)^{3}+\frac{\partial z}{\partial t} =0
\end{equation*}
is a first - order equation in two variables, while
\begin{equation*}
\frac{\partial^{2} z}{\partial x^{2}}=\frac{\partial z}{\partial t}
\end{equation*}
and
\begin{equation*}
\frac{\partial^{2} z}{\partial x^{2}}+2\frac{\partial^{2} z}{\partial x \partial y}+\frac{\partial^{2} z}{\partial y^{2}}=0
\end{equation*}
are a second order eqution in two variables. The coefficients of derivaritives
\begin{equation*}
\frac{\partial z}{\partial x}, \frac{\partial^{2} z}{\partial x^{2}}, \frac{\partial^{2} z}{\partial y^{2}}, \frac{\partial^{2} z}{\partial x \partial y}, \frac{\partial z}{\partial t}
\end{equation*}
etc. are called the partial differential coefficients.
PDEs can be classified into different types based on their characteristics. Some common types of PDEs include: Elliptic PDEs: These equations involve second-order derivatives and typically arise in steady-state problems. Examples include Laplace’s equation and Poisson’s equation. Parabolic PDEs: These equations involve a first-order time derivative and second-order spatial derivatives. They often describe phenomena that evolve over time. The heat equation and the diffusion equation are examples of parabolic PDEs. Hyperbolic PDEs: These equations involve second-order time derivatives and second-order spatial derivatives. They are often used to describe wave-like phenomena. The wave equation and the telegraph equation are examples of hyperbolic PDEs.
The following equations are partial differential equations that are generally appeared in the problems of physics.
- wave equation:\begin{equation*} \nabla^{2}\psi = \frac{1}{v^{2}}\frac{\partial^{2} \psi}{\partial t^{2}}, \end{equation*}
- Diffusion (heat) equation:\begin{equation*} \nabla^{2}\psi = \frac{1}{h^{2}}\frac{\partial \psi}{\partial t}, \quad \text{and} \end{equation*}
- Laplace’s equation:\begin{equation*} \nabla^{2}\psi =0 \end{equation*}
In this chaptor we shall try to find their solutions by using a very powerful method, known as a method of separation of variables. In this method the dependent variable is assumed to be the product of two functions, each of which involves only one of the independent variables and forms two ordinary differential equations.
For python code of PDE, go to
https://docs.sympy.org/latest/modules/solvers/pde.html
