4.6 Classification and Limitations

4.6 Classification and Limitations

• We have seen many equations and solutions by this time.

• Wave Oscillatory is not always periodic.

• The three two-variable equations are representative of the three classes of second-order linear partial differential equations.

• The potential equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic.
• When numerical techniques are used for solution, the method of attack is determined by the classification of an equation.

• Does separation of variables work on all equations?
• The answer is no.

• It is difficult to say which equations can be solved by this method.

• The region in which the solution is to be found is limiting the applicability of the method we have used.
• A region is defined by coordinate curves of the partial differential equation.
• The region is described by inequalities on the coordinates.

• All of these are rectangular, but only one is an ordinary one.

• If we applied our methods to the potential equation in the L-shaped region, they would break down.

• There are restrictions on the kinds of boundary conditions that can be dealt with.

• If two or more functions satisfy the conditions, so does a sum of those functions.

• Despite the limitations of the method of separation of variables, it works well on many important problems in two or more variables and provides insight into the nature of their solutions.
• It is known that if there is a separation of variables, it will find a solution.

• Classify the equations.

• The text has regions listed as generalized rectangles.

• Evaluate the solutions to the three problems.