PARTIAL DIFFERENTIAL EQUATION

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multi variable functions and their partial derivatives. ... A special case is ordinary differential equations (ODEs), which deal with functions of a single variable and their derivatives.

Partial differential equations (PDEs) have just one small change from ordinary differential equations - but it makes it significantly harder. In general the vast majority cannot be solved analytically. But a small class of special partial differential equations can be solved analytically.An ordinary differential equation (ODE) is an equation that involves some ordinary derivatives (as opposed to partial derivatives) of a function. Often, our goal is to solve an ODE, i.e., determine what function or functions satisfy the equation

after a brief conversation on calculus,for those that hunger for progression,surviving Calculus 1,2 and 3...deferential equations await,still want to be a physicist?solving PDE....first....separate the variables,Multiply both sides by dx:dy = (1/y) dx. Multiply both sides by y: y dy = dx.

1. Put the integral sign in front:∫ y dy = ∫ dx. Integrate each side: (y2)/2 = x + C.
2. Multiply both sides by 2: y2 = 2(x + C)In mathematics, separation of variables (also known as the Fourier method) is any of several methods for solving ordinary and partial differential equations, in which algebra allows one to rewrite an equation so that each of two variables occurs on a different side of the equation,who is to blame for this mathematical stress.....Gottfried Wilhelm Leibniz,PDEs can be used to describe a wide variety of phenomena such as sound, heat, diffusion, electrostatics, electrodynamics, fluid dynamics, elasticity, gravitation and quantum mechanics. These seemingly distinct physical phenomena can be formalised similarly in terms of PDEs. Just as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. PDEs find their generalisation in stochastic partial differential equations.An integral transform may transform the PDE to a simpler one, in particular, a separable PDE. This corresponds to diagonalizing an operator.

An important example of this is Fourier analysis, which diagonalizes the heat equation using the eigenbasis of sinusoidal waves.

If the domain is finite or periodic, an infinite sum of solutions such as a Fourier series is appropriate, but an integral of solutions such as a Fourier integral is generally required for infinite domains.