Find a solution for differential equations

In summary, the constants A,B,L,M for the solution to the differential equation can only be found if the function h is continuous at r_0.
  • #1
ip88
3
0
I have to solve the following differential equation (that I found in an article):

[itex]\frac{1}{r^{5}}\partial_{r}(r^{5}\partial_{r}h(r)) -E\frac{h(r)}{r^{2}} = - \frac{C}{r^{5}}\delta(r-r_{0})[/itex]

where E and C are two constants.

The authors of the article first find a solution of the previous equation when [itex]r≠r_{0}[/itex] and the result is:

[itex]h(r)=Ar^{c_{\pm}}[/itex] where [itex]c_{\pm} = -2 \pm \sqrt{E+4} [/itex]

Then they have to find the value of the constant A. In order to reach this aim they say that I have to integrate the differential equation and the result should be:

[itex]\frac{C}{2\sqrt{E+4}}\times [/itex] [itex]\frac{1}{r_{0}^{4}}(\frac{r}{r_{0}})^{c_{+}}[/itex] if r ≤ r_{0}

and

[itex]\frac{C}{2\sqrt{E+4}}\times [/itex] [itex]\frac{1}{r^{4}}(\frac{r_{0}}{r})^{c_{+}}[/itex] if r ≥ r_{0}

However I'm not able to recover this result. The point is that when I insert the solution of the homogeneous equation and I integrate the left side of the equation is always equal to zero. Probably I did not understand correctly what is the procedure that I have to follow in order to find the solution to the inhomogeneous equation (once I have the solution of the homogeneous equation), but I do not know how to reach this aim.
 
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  • #2
The general solution for [itex]r \neq r_0[/itex] is
[tex]h(r)=A_1 r^{c_+}+A_2 r^{c_-},[/tex]
where the constants [itex]A_1[/itex] and [itex]A_2[/itex] can be different for [itex]r<r_0[/itex] and [itex]r>r_0[/itex] in such a way that you get the [itex]\delta[/itex]-distribution singularity. Since it's a 2nd-order equation you can demand that [itex]h[/itex] is continuous at [itex]r_0[/itex]. This gives you one constraint for your four unknowns. You get a 2nd constraint by integrating the differential equation over an infinitesimal interval containing [itex]r_0[/itex] in its interior, leading to a condition for the appropriate jump in [itex]h'(r)[/itex].

Further you need some more conditions to fix the function completely. These should be given from the application treated with this function (Green's function for some physical problem?).
 
  • #3
Hi ip88,

To expand upon what vanhees71 said.

After solving the homgenous equation you have two different solutions


[itex] h_< = Ar^{c_+} + B r^c{_-}[/itex]
and
[itex] h_> = L r^{c_+} + M r^{c_-}[/itex]

The solution [itex] h_< [/itex] is only valid for [itex] r < r_0[/itex] and [itex] h_>[/itex] is only valid for [itex] r > r_0[/itex].

The goal is then to find the constants [itex]A,B,L, M[/itex]. There are 4 unknowns, and we need 4 equations. Two equations come from the boundary conditions of the ODE. A third equation is obtained by requiring [itex] h_< =h_> [/itex] at [itex] r_0 [/itex].

To get a forth equation we integrate the differential equation from [itex] r_0 -\epsilon [/itex] to [itex] r_0 +\epsilon [/itex] and then take the limit that [itex] \epsilon \rightarrow 0 [/itex]. This last step is where all the magic happen and it gives you a "jump condition".

To see how this works let's take the integral: (I've also multiplied the equation by r^5)
[itex] \int_{r_0-\epsilon}^{r_0+\epsilon} dr \left(\partial_r (r^5 \partial_r h) - Er^3h=-C\delta(r-r0) \right) [/itex]

The integral of the first term is:
[itex] \left. r^5 \partial_r h \right|_{r_0-\epsilon}^{r_0+\epsilon} [/itex]
and in the limit that [itex] \epsilon \rightarrow 0 [/itex] we get [itex] r_0^5 \left(h'_>-h'_< \right)\left. \right|_{r_0} [/itex]. The primes are the derivatives in r.

The integral of the second term goes to zero in the limit that [itex] \epsilon \rightarrow 0 [/itex].

Finally the integral of the third term is -C

Putting this together gives the jump condition
[itex] r_0^5 \left(h'_>-h'_< \right)\left. \right|_{r_0} =-C[/itex]
 
  • #4
Thank you very much for the explanation
 
  • #5


As a scientist, it is important to carefully follow the steps and procedures outlined in any article or research paper. In this case, the authors have provided a solution for the given differential equation when r≠r_{0}. However, it is understandable that you are having difficulty in understanding the integration process to find the value of the constant A.

One possible explanation for the discrepancy could be that the authors have made a mistake in their integration process. It is important to carefully check all the steps and assumptions made during the integration process to ensure accuracy.

Another possibility could be that the authors have not clearly explained the integration process and the steps required to find the value of A. In this case, it is important to reach out to the authors for clarification or seek help from a colleague who is familiar with the topic.

In any case, as a scientist, it is important to critically evaluate and verify the results presented in any research paper. If you are unable to replicate the results or find discrepancies, it is important to communicate this to the authors and seek clarification. This will not only help in improving the accuracy of the research, but also contribute to the advancement of science.
 

1. How can I solve a differential equation?

To solve a differential equation, you will need to use mathematical techniques such as integration, differentiation, substitution, and separation of variables. It is also important to have a good understanding of the properties of differential equations and the specific type of equation you are trying to solve.

2. Can I use software to find a solution for a differential equation?

Yes, there are many software programs available that can help you find a solution for a differential equation. Some popular options include Mathematica, MATLAB, and Maple. However, it is still important to have a good understanding of the mathematical concepts involved in solving differential equations.

3. Are there different methods for solving differential equations?

Yes, there are several methods for solving differential equations, including Euler's method, separation of variables, and power series. The method you use will depend on the type of differential equation and the initial conditions given.

4. How do I know if my solution for a differential equation is correct?

You can check the validity of your solution by plugging it back into the original differential equation. If it satisfies the equation, then it is a valid solution. You can also compare your solution to known solutions or use a graphing calculator to visualize the solution.

5. Can I solve any differential equation?

While it is possible to solve many differential equations, some equations may be too complex or have no known analytical solution. In these cases, numerical methods or approximations may be used to find an approximate solution. Additionally, some differential equations may have multiple solutions or no solution at all.

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