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Partial Differential Equations

A partial differential equation is a differential equation which contains unknown multivariable functions and their partial derivatives. They are used to solve problems involving functions of several variables. They can usually be solved by hand however occasionally a computer is necessary. In physics problems, partial differential equations are used to describe a variety of phenomena such as electrostatics, fluid flow, sound, heat and elasticity.

Partial differential equations are problems that involve rates of change with respect to a continuous variable. The equation for a partial differential equation function is:

F(x_1,……,x_n,u,∂u/∂x_1 ,……,∂u/(∂x_n ),(∂^2 u)/(∂x_1 ∂x_2 ),……,(∂^2 u)/(∂x_1 ∂x_n ),…)=0

Where F is a linear function of u and its derivatives.

Partial differential equations can be solved using Laplace transforms, numerical methods or on a computer. The method depends on the order of the equation. Therefore, partial differential equations are extremely useful when dealing with single order or multi-variable systems which occur very often in physics problems.

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