How do differential equations work

This makes sense, because the spring is not exerting a force at that moment. On the other hand, when the position is large ie. The task is to find a function whose various derivatives fit the differential equation over a long span of time. For example,. It is easy to confirm that you have a solution: just plug the solution in to the differential equation! Usually, we'll skip many of these steps and use a shortcut method. Let's revisit our solution method to see how we can take some shortcuts.

Next, observe the results of the substitution. The simple ODEs of this introduction give you a taste of what ordinary differential equations are and how we can solve them. You can check out some examples involving equations that you can solve just with the techniques learned here. Home Threads Index About. An introduction to ordinary differential equations. What are ordinary differential equations ODEs?

We can note two additional points about these values:. Finally, some descriptive comments about the details of the program in Appendix 1 are given immediately after the program listing. We again have an analytical solution to evaluate the numerical solution. Since Eq. An important difference between the parabolic problem of Eq. Equations 66 , 67 , 68 , 69 , 70 , 71 constitute the complete finite difference approximation of Eqs. A small MATLAB program for this approximation is given in Appendix 2 , including some descriptive comments immediately after the program.

The general analytical solution to Eq. The plotted output from the program is given in Figure 1 and includes both the numerical solution of Eqs. We can note the following points about Figure 1 :. The parameters that produced the numerical output in Figure 1 are listed in Table 3. Additional parameters follow from the values in Table 3.

This is a stringent test of the numerical solution since the curvature of the solution is greatest at these peaks. The results are summarized in Table 4. Of course, the peak analytical values given by Eq. This explicit finite difference numerical solution also has a stability limit like the preceding parabolic problem.

In other words, the parameters of Table 3 were chosen primarily for accuracy and not stability. Rather, we will convert Eq. The idea then is to integrate Eq. The analytical solution to Eqs. The analytical solution of Eq. To develop a numerical solution to Eq. The output from this program is listed in Table 5. The convergence of the solution of Eq. The parameters that produced the numerical output in Table 5 are listed in Table 6.

Additional parameters follow from the values in Table 6. The stability constraint for the 2D problem of Eq. The actual path that the parabolic problem takes to the solution of the elliptic problem is not relevant so long as the parabolic solution converges to the elliptic solution. Some other trial values indicated that this initial value is not critical but it should be as close to the final value, for example 2. This parametrization is an example of continuation in which the solution is continued from the given assumed starting value of Eq.

The concept of continuation has been applied in many forms and not just through the addition of a derivative as in Eq. In general, the errors in the numerical solution of PDEs can result from the limited accuracy of all of the approximations used in the calculation.

For example, the 0. In formulating a numerical method or algorithm for the solution of a PDE problem, it is necessary to balance the discretization errors so that one source of error does not dominate, and generally degrade, the numerical solution.

Thus, control of approximation errors is central to the calculation of a numerical solution of acceptable accuracy. In the preceding examples, this control of errors can be accomplished in three ways:. The three preceding numerical solutions were developed using basic finite differences such as in Eqs. However, many approaches to approximating derivatives in PDEs have been developed and used.

Among these are finite elements, finite volumes, weighted residuals , e. Each of these methods has advantages and disadvantages, often according to the characteristics of the problem of interest starting with the parabolic, hyperbolic and elliptic geometric classifications. Thus, an extensive literature for the numerical solution of PDEs is available, and we have only presented here a few basic concepts and examples.

The principal advantage of numerical methods applied to PDEs is that, in principle, PDEs of any number and complexity can be solved which is particularly useful when analytical solutions are not available. As another example, a solution to the Burgers equation could be computed by extending Eq. While Euler's method is general with respect to the form of the initial value integration, it does have two important limitations:.

Thus, the Euler method is limited by both accuracy and stability. As might be expected, such higher order methods are more complicated than the Euler method, but fortunately, they have been programmed in library routines that can easily be called and used.

The use of library routines for initial value integration is the basis for much of the work in the numerical method of lines solution of PDEs. This rule of thumb is : Start with real numbers, end with real numbers. So, we saw in the last example that even though a function may symbolically satisfy a differential equation, because of certain restrictions brought about by the solution we cannot use all values of the independent variable and hence, must make a restriction on the independent variable.

This will be the case with many solutions to differential equations. In the last example, note that there are in fact many more possible solutions to the differential equation given. For instance, all of the following are also solutions. Given these examples can you come up with any other solutions to the differential equation? There are in fact an infinite number of solutions to this differential equation. So, given that there are an infinite number of solutions to the differential equation in the last example provided you believe us when we say that anyway….

Which is the solution that we want or does it matter which solution we use? This question leads us to the next definition in this section. Initial Condition s are a condition, or set of conditions, on the solution that will allow us to determine which solution that we are after.

Initial conditions often abbreviated i.