Solution to Differential Equations Using Discrete Green's Function and Duhamel's Methods
This Demonstration shows how to find approximate solutions to linear ordinary differential equations using two methods:
1. the discrete Green's function method, in which the source is approximated as a sequence of pulses;
2. the discrete Duhamel's method, in which the source is approximated by a sequence of strips.
The complete solution is approximated by a superposition of solutions for each individual pulse or strip. As the limit of the number of segments tends to infinity, the pulse and strip methods approach the continuous Green's method and Duhamel's method, respectively.
In the graphs:
• solid, filled, red lines represent the exact source and response;
• thin, black, dashed lines represent the individual sources and responses to each pulse or strip;
• thick, black, dashed lines represent the total approximate source and response.
Linear problems can be constructed by using a linear combination from simpler subproblems. If the forcing function is expressed as , then the total solution can be expressed as , where is the response to . In this Demonstration, we have decomposed the forcing function, , as a sequence of either pulses or strips. This method can be extended to solve problems that contain other types of forcing functions such as volumetric sources, surface sources, and initial conditions. The approximate solution with multiple forcing functions can be found by superimposing each response to the individual forcing functions.
The pulse method uses the linearity property to construct the total response by the addition or superposition of the responses to all the pulses. This method is a discrete version of the Green's function technique. In the limit as , the method reduces to the usual Green's function method. Similarly, the strip method finds the total response by the superpostion of responses to all the strips. This method is particularly useful in the time variable rather than the spatial variable, because time runs indefinitely, while space is typically limited to a finite region. This method is a discrete analog of Duhamel's method. By taking the limit as , Duhamel's method is obtained. Each method has its advantages and disadvantages, but they result in the same level of approximation.