Elliptic and parabolic equations are used for modeling:
Steady and unsteady heat transfer in solids
Flows in porous media and diffusion problems
Electrostatics of dielectric and conductive media
Steady state of wave equations
Hyperbolic equation is used for:
Transient and harmonic wave propagation in acoustics and electromagnetics
Transverse motions of membranes
Eigenvalue problems are used for:
Determining natural vibration states in membranes and structural mechanics problems
In addition to solving generic scalar PDEs and generic systems of PDEs with vector valued u, Partial Differential Equation Toolbox™ provides tools for solving PDEs that occur in these common applications in engineering and science:
The PDE app lets you specify PDE coefficients and boundary conditions in terms of physical entities. For example, you can specify Young's modulus in structural mechanics problems.
The application mode can be selected directly from the pop-up menu in the upper right part of the PDE app or by selecting an application from the Application submenu in the Options menu. Changing the application resets all PDE coefficients and boundary conditions to the default values for that specific application mode.
When using an application mode, the generic PDE coefficients are replaced by application-specific parameters such as Young's modulus for problems in structural mechanics. The application-specific parameters are entered by selecting Parameters from the PDE menu or by clicking the PDE button. You can also access the PDE parameters by double-clicking a subdomain, if you are in the PDE mode. That way it is possible to define PDE parameters for problems with regions of different material properties. The Boundary condition dialog box is also altered so that the Description column reflects the physical meaning of the different boundary condition coefficients. Finally, the Plot Selection dialog box allows you to visualize the relevant physical variables for the selected application.
The PDE app lets you solve problems with vector valued u of dimension two. However, you can use functions to solve problems for any dimension of u.