Stuck with a solution for LSE

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In summary: Your Name]In summary, the conversation was about solving the Eikonal equation using the method of characteristics. This involves finding a set of curves along which the solution is constant, and then transforming the equation into a system of ODEs. The solution can then be obtained by solving these ODEs and plugging the resulting functions back into the original equation.
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Hi,
I am looking for a solution of

[itex]\frac{\partial\phi}{\partial t} = |\nabla\phi|div\left(\frac{\nabla\phi}{|\nabla\phi|}\right)[/itex]

in the form of

[itex]\phi (x_1,x_2,t)=\gamma (t)\sqrt{x_1^2+x_2^2}-\alpha (t)[/itex]

i got myself to, where i want to evaluate [itex]\gamma, \alpha[/itex]

[itex]\dot{\gamma}(t)\sqrt{x_1^2+x_2^2} - \dot{\alpha}(t)=\frac{1}{\sqrt{x_1^2+x_2^2}}.[/itex]

Anyone having an idea how to do this, or am i missing something? It's been a while since i last used PDE so this is as far as i got.
any kind of help is appreciated
 
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Hello,

Thank you for reaching out with your question. The equation you are trying to solve is a nonlinear partial differential equation known as the Eikonal equation. This equation is commonly used in geometric optics and can also be seen in other areas of physics and mathematics.

To solve this equation, you can use the method of characteristics. This method involves finding a set of curves, known as characteristics, along which the solution to the equation is constant. These curves can then be used to transform the original equation into a set of ordinary differential equations, which can be solved using standard techniques.

In your case, the characteristics are given by the equation \frac{dx_1}{dt} = |\nabla\phi| and \frac{dx_2}{dt} = |\nabla\phi|. Using these equations, you can transform the original equation into the following system of ODEs:

\dot{\gamma}(t) = |\nabla\phi| and \dot{\alpha}(t) = -\frac{1}{2}|\nabla\phi|^2.

These equations can be solved using standard techniques, such as separation of variables or numerical methods, to obtain the functions \gamma(t) and \alpha(t). Once you have these functions, you can plug them back into the original equation to get the solution \phi(x_1,x_2,t).

I hope this helps you in solving your problem. If you have any further questions, please don't hesitate to ask. Best of luck with your research.
 

What is LSE?

LSE stands for London School of Economics and Political Science. It is a prestigious university in London known for its social science programs.

What does it mean to be "stuck with a solution for LSE"?

Being "stuck with a solution for LSE" refers to being unable to find a satisfactory solution to a problem or challenge related to the university. This could include issues with coursework, research, or other aspects of academic life.

What are some common problems that students may face at LSE?

Some common problems that students may face at LSE include high levels of academic pressure, difficulty balancing coursework with extracurricular activities, and challenges with adjusting to a new academic and cultural environment.

How can one find a solution for LSE-related issues?

There are several ways to find solutions for LSE-related issues. These include seeking support from academic advisors, reaching out to peers for guidance, and utilizing resources such as tutoring or counseling services offered by the university.

Are there any resources specifically designed for students struggling with solutions at LSE?

Yes, LSE offers a variety of resources for students struggling with solutions, including academic support services, peer mentoring programs, and workshops on time management and study skills. Students can also seek assistance from their professors or reach out to the student union for support.

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