Essay On Partial Differential Equation

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ABSTRACT

A partial differential equation is a differential equation that contain unknown multivariable functions and their partial derivatives while ordinary differential equations contains function of a single variables and their derivatives. Therefore, an ordinary differential equation is a special case of partial differential equation but the behaviour of a solution is quite different. It is much more complicated in the case of partial differential equation because it has more than one independent variable. Partial differential equation can be used to describe the various topics in physics such as propagation of sound and wave, heat, electrostatics, Maxwell equation in electromagnetism, elasticity, quantum mechanics, Hamilton-Jacobi theory. …show more content…

The independent variables will be denoted by x and y and the dependent variables denoted by z. The partial differentials coefficients are denoted as follows:
∂z/∂x=p ∂z/∂y=q (∂^2 z)/〖∂x〗^2 =r (∂^2 z)/∂x∂y=s (∂^2 z)/〖∂y〗^2 …show more content…

(∂^2 V)/〖∂x〗^2 =LC(∂^2 V)/〖∂t〗^2
(∂^2 i)/〖∂x〗^2 =LC(∂^2 i)/〖∂t〗^2
These equations are called radio equations.
Where,
V=potential i=current C=capacitance L inductance

IN HEAT FLOW[8]
Heat is the transfer of energy between two bodies. One dimensional heat equation
When the heat flow along the straight lines, then the flow is called one-dimensional.
Let us consider the heat flow in a bar or rod along the x-axis. Consider a rod of homogeneous material of density is ρ (gm⁄〖cm〗^3 ) and having a constant cross-sectional area A (〖cm〗^2 ). Let c be the specific heat and k be thermal conductivity of material. We suppose that the sides of the bar are insulated and the loss of heat from the sides by conduction or radiations is negligible. Take an end of the bar as the origin and the direction of heat flow as the positive x-axis.

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