Replacing a non-harmonic function with a harmonic function

[shahab44]

TL;DR Summary

replacing a non-harmonic function with a harmonic function

I am solving a problem of the boundary condition of Dirichlet type, in order to solve the problem, the functions within the differential equations suppose to be harmonic.

I have a function f(x,y,z) (the function attached) which is not harmonic. I must find an equivalent function g(x,y,z) which shall its Laplacian to be zero and at the boundary which is x=0 to be equal to f(x,y,z) "i.e f(0,y,z)=g(0,y,z)"
I have been trying with Mathematica for almost a week but just by trail, which is not a clever way. I am wondering if there is a way to do it.

thanks

 

[jvicens]

Thank you for sharing your problem with the community. I can understand the frustration of trying to solve a difficult problem without a clear solution in sight. However, I believe there are a few steps you can take to approach this problem in a more systematic and efficient manner.

Firstly, it is important to understand the concept of a harmonic function and its relationship with the Laplace equation. A function is considered harmonic if it satisfies the Laplace equation, which is a second-order partial differential equation. This equation states that the sum of the second-order partial derivatives of a function with respect to its variables must equal zero. In other words, a function is harmonic if its Laplacian (the sum of its second-order partial derivatives) is equal to zero.

In your case, you have a function f(x,y,z) that is not harmonic. This means that its Laplacian is not equal to zero. In order to find an equivalent function g(x,y,z) that is harmonic, you will need to manipulate f(x,y,z) in a way that its Laplacian becomes zero. This can be achieved by adding a term or terms to f(x,y,z) that cancel out the non-zero terms in its Laplacian.

To do this, you can start by considering the Taylor series expansion of f(x,y,z) around the point x=0. This will give you an expression for f(x,y,z) in terms of its derivatives at x=0. From there, you can manipulate this expression to find the additional terms that need to be added to f(x,y,z) in order for its Laplacian to be equal to zero. This approach will help you find a more systematic and analytical solution, rather than relying on trial and error.

Additionally, you can also consult with experts in the field of differential equations or seek help from colleagues who may have experience with similar problems. Collaborating with others can often lead to new insights and approaches that you may not have considered before.

I hope this helps and I wish you the best of luck in solving this problem.