Partial Differential Equation in Matlab Programming

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A partial differential equation (PDE) is a type of differential equation that contains before-hand unknown multivariable functions and their partial derivatives. PDEs are used to make problems involving functions of several variables, and are either solved by hand, or used to create a computer model. A special case of this is ordinary differential equations (ODEs), which only works with functions of a single variable and their derivatives. PDEs can be used to define a large variety of phenomena such as sound, heat, electrostatics, fluid dynamics, electrodynamics, elasticity, or quantum mechanics. These different physical phenomena can be transformed similarly in terms of PDEs. Like as ordinary differential equations often model one-dimensional dynamical systems, partial differential equations often model multidimensional systems. For solving partial differential equation using MATLAB modelling involves Basically the two functions that are available in MATLAB that help in solving partial differential equations.

• • PDE Solver function

• • PDE Helper Function

PDE Solver function

The MATLAB PDE solver function, pdepe, solves the initial-boundary value problems for systems of parabolic and elliptic PDEs in the one space variable x and time t. There should be at least one parabolic equation in the system.

The pdepe solver transforms the PDEs to ODEs using a second-order accurate spatial discretization based on a fixed set of user-specified nodes. The discretization method is described as The time integration is done with ode15s. The pdepe solver makes full use of the capabilities of ode15s for solving the differential-algebraic equations.

The basic syntax of the solver is:

sol = pdepe(m,pdefun,icfun,bcfun,xmesh,tspan)

PDE Helper Function

This function in MATLAB computes the numerical solution of PDE with the help of output of pdepe

[uout,duoutdx] = pdeval(m,x,ui,xout)

[uout,duoutdx] = pdeval(m,x,ui,xout) approximates the solution ui and its partial derivative ∂ui/∂x at points from the interval [x0,xn]. The pdeval function results the calculated values in uout and duoutdx, respectively.