Fortunately, partial differential equations of second-order are often amenable to analytical solution. Such PDEs are of the form. Linear second-order PDEs are then classified according to the properties of the matrix.
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If is a positive definite matrix , i. Laplace's equation and Poisson's equation are examples. Boundary conditions are used to give the constraint on , where.
Basic linear partial differential equations, Volume 62
If det , the PDE is said to be hyperbolic. The wave equation is an example of a hyperbolic partial differential equation. Initial-boundary conditions are used to give. If det , the PDE is said to be parabolic.
The heat conduction equation equation and other diffusion equations are examples. The following are examples of important partial differential equations that commonly arise in problems of mathematical physics. Spherical harmonic differential equation. Arfken, G. Orlando, FL: Academic Press, pp. Bateman, H. Partial Differential Equations of Mathematical Physics. New York: Dover, Conte, R.
Kamke, E. New York: Chelsea, Folland, G. Introduction to Partial Differential Equations, 2nd ed. Kevorkian, J.
Elements of Partial Differential Equations
New York: Springer-Verlag, Morse, P. New York: McGraw-Hill, pp. Polyanin, A. Separation of Variables — In this section show how the method of Separation of Variables can be applied to a partial differential equation to reduce the partial differential equation down to two ordinary differential equations.
We apply the method to several partial differential equations. We do not, however, go any farther in the solution process for the partial differential equations. That will be done in later sections.
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The point of this section is only to illustrate how the method works. Solving the Heat Equation — In this section we go through the complete separation of variables process, including solving the two ordinary differential equations the process generates. We will do this by solving the heat equation with three different sets of boundary conditions. Heat Equation with Non-Zero Temperature Boundaries — In this section we take a quick look at solving the heat equation in which the boundary conditions are fixed, non-zero temperature. Note that this is in contrast to the previous section when we generally required the boundary conditions to be both fixed and zero.
As we will see this is exactly the equation we would need to solve if we were looking to find the equilibrium solution i. Vibrating String — In this section we solve the one dimensional wave equation to get the displacement of a vibrating string. Summary of Separation of Variables — In this final section we give a quick summary of the method of separation of variables for solving partial differential equations.
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- Linear Partial Differential Equations of Mathematical Physics.
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