General question of Lie groups results
I've been trying to self study Lie groups and right now i have a question about the general theory, so far i have seen that
For example, in Peter's book (Applications of Lie Groups to Differential equations) there is an example with the heat equation
$$u_t=u_{xx}$$
after applying the theory, it says; Thus the most general solution obtainable from a given solution $u=f(x,t)$ by group transformations is of the form
$$u=frac{1}{sqrt{1+4a_6}}text{exp}left{a_3-frac{a_5x+a_6x^2-a_5^2t}{1+4e_6t}right}times fleft(frac{e^{e_4}(x-2e_5t)}{1+4a_6t}-a_1,frac{e^{2a_4}t}{1+4a_6t}-a_2right)+alpha(x,t),$$
where $a_1,ldots,a_6$ are real constants and $alpha$ an arbitrary solution to the heat equation.
- So my question is, what does this tells us about the differential equation? what bothers me its the fact that I am thinking or i want to get an exact solution for the heat equation, then i started to remember from a course i had in ordinal differential equations, when you can't find an exact solution, but you found properties of the equations that let you have an idea of how the solutions behave, so, maybe Lie groups gives us similar information about the equations, but I am a little lost in how to see this results, any help would be appreciated, thanks.
lie-groups
asked Nov 28 '18 at 2:37
Zigisfredo
308111
add a comment |
I've been trying to self study Lie groups and right now i have a question about the general theory, so far i have seen that
For example, in Peter's book (Applications of Lie Groups to Differential equations) there is an example with the heat equation
$$u_t=u_{xx}$$
after applying the theory, it says; Thus the most general solution obtainable from a given solution $u=f(x,t)$ by group transformations is of the form
$$u=frac{1}{sqrt{1+4a_6}}text{exp}left{a_3-frac{a_5x+a_6x^2-a_5^2t}{1+4e_6t}right}times fleft(frac{e^{e_4}(x-2e_5t)}{1+4a_6t}-a_1,frac{e^{2a_4}t}{1+4a_6t}-a_2right)+alpha(x,t),$$
where $a_1,ldots,a_6$ are real constants and $alpha$ an arbitrary solution to the heat equation.
- So my question is, what does this tells us about the differential equation? what bothers me its the fact that I am thinking or i want to get an exact solution for the heat equation, then i started to remember from a course i had in ordinal differential equations, when you can't find an exact solution, but you found properties of the equations that let you have an idea of how the solutions behave, so, maybe Lie groups gives us similar information about the equations, but I am a little lost in how to see this results, any help would be appreciated, thanks.
lie-groups
asked Nov 28 '18 at 2:37
Zigisfredo
308111
add a comment |
I've been trying to self study Lie groups and right now i have a question about the general theory, so far i have seen that
For example, in Peter's book (Applications of Lie Groups to Differential equations) there is an example with the heat equation
$$u_t=u_{xx}$$
after applying the theory, it says; Thus the most general solution obtainable from a given solution $u=f(x,t)$ by group transformations is of the form
$$u=frac{1}{sqrt{1+4a_6}}text{exp}left{a_3-frac{a_5x+a_6x^2-a_5^2t}{1+4e_6t}right}times fleft(frac{e^{e_4}(x-2e_5t)}{1+4a_6t}-a_1,frac{e^{2a_4}t}{1+4a_6t}-a_2right)+alpha(x,t),$$
where $a_1,ldots,a_6$ are real constants and $alpha$ an arbitrary solution to the heat equation.
- So my question is, what does this tells us about the differential equation? what bothers me its the fact that I am thinking or i want to get an exact solution for the heat equation, then i started to remember from a course i had in ordinal differential equations, when you can't find an exact solution, but you found properties of the equations that let you have an idea of how the solutions behave, so, maybe Lie groups gives us similar information about the equations, but I am a little lost in how to see this results, any help would be appreciated, thanks.
lie-groups
asked Nov 28 '18 at 2:37
Zigisfredo
308111
I've been trying to self study Lie groups and right now i have a question about the general theory, so far i have seen that
For example, in Peter's book (Applications of Lie Groups to Differential equations) there is an example with the heat equation
$$u_t=u_{xx}$$
after applying the theory, it says; Thus the most general solution obtainable from a given solution $u=f(x,t)$ by group transformations is of the form
$$u=frac{1}{sqrt{1+4a_6}}text{exp}left{a_3-frac{a_5x+a_6x^2-a_5^2t}{1+4e_6t}right}times fleft(frac{e^{e_4}(x-2e_5t)}{1+4a_6t}-a_1,frac{e^{2a_4}t}{1+4a_6t}-a_2right)+alpha(x,t),$$
where $a_1,ldots,a_6$ are real constants and $alpha$ an arbitrary solution to the heat equation.
- So my question is, what does this tells us about the differential equation? what bothers me its the fact that I am thinking or i want to get an exact solution for the heat equation, then i started to remember from a course i had in ordinal differential equations, when you can't find an exact solution, but you found properties of the equations that let you have an idea of how the solutions behave, so, maybe Lie groups gives us similar information about the equations, but I am a little lost in how to see this results, any help would be appreciated, thanks.
lie-groups
lie-groups
asked Nov 28 '18 at 2:37
Zigisfredo
308111
asked Nov 28 '18 at 2:37
Zigisfredo
308111
asked Nov 28 '18 at 2:37
Zigisfredo
308111
asked Nov 28 '18 at 2:37
Zigisfredo
308111
asked Nov 28 '18 at 2:37
Zigisfredo
308111
308111
add a comment |
add a comment |
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