What type of PDE's can one solve using Fourier Transforms? The Next CEO of Stack OverflowUsing games to approximate solutions to PDE'sHeat equation, separation of variables and Fourier transformPartial differential equation (heat equation with other terms)?Rewriting the heat diffusion equation with temperature dependent diffusion coefficient to include joule heating.Good recommendations for solving PDE's by integral transformsFourier transform - next stepSolving Laplace Equation with Fourier IntegralSolving the Heat Equation using the Fourier TransformMotivation on Using Fourier Series to Solve Heat EquationSolving an integral equation (possibly Fredholm, 1st kind) containing quartic exponentials with Fourier Transforms
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What type of PDE's can one solve using Fourier Transforms?
The Next CEO of Stack OverflowUsing games to approximate solutions to PDE'sHeat equation, separation of variables and Fourier transformPartial differential equation (heat equation with other terms)?Rewriting the heat diffusion equation with temperature dependent diffusion coefficient to include joule heating.Good recommendations for solving PDE's by integral transformsFourier transform - next stepSolving Laplace Equation with Fourier IntegralSolving the Heat Equation using the Fourier TransformMotivation on Using Fourier Series to Solve Heat EquationSolving an integral equation (possibly Fredholm, 1st kind) containing quartic exponentials with Fourier Transforms
2
$begingroup$
The question is really in the title. I have been seeing many examples of PDE's (heat equation on an infinite domain for example) being solved using Fourier transforms (FT). However, I have been unable to find a theory that says which type of PDE's can be solved by FT. So, my questions would be:
- What does mathematical theory say regarding the type of PDE's that can be solved by FT?
- Is the following PDE solvable by FT : $c cdot varphi_EE + f(E) cdot varphi_E + g(E) cdot varphi + varphi_x = 0 $, where $varphi = varphi(E,x)$, $E in (-infty,+infty)$, $ x in (0,infty)$, $varphi_EE=fracpartial^2varphipartial E^2$ and $varphi_x=fracpartialvarphipartial x$ and $c = constant$
- If the above equation is not solvable by Fourier Transforms, what other methods (aside from numerical solutions) are applicable?
Thank you in advance
pde fourier-transform
asked yesterday
TiberiuTiberiu
161
New contributor
Tiberiu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
add a comment |
2
$begingroup$
The question is really in the title. I have been seeing many examples of PDE's (heat equation on an infinite domain for example) being solved using Fourier transforms (FT). However, I have been unable to find a theory that says which type of PDE's can be solved by FT. So, my questions would be:
- What does mathematical theory say regarding the type of PDE's that can be solved by FT?
- Is the following PDE solvable by FT : $c cdot varphi_EE + f(E) cdot varphi_E + g(E) cdot varphi + varphi_x = 0 $, where $varphi = varphi(E,x)$, $E in (-infty,+infty)$, $ x in (0,infty)$, $varphi_EE=fracpartial^2varphipartial E^2$ and $varphi_x=fracpartialvarphipartial x$ and $c = constant$
- If the above equation is not solvable by Fourier Transforms, what other methods (aside from numerical solutions) are applicable?
Thank you in advance
pde fourier-transform
asked yesterday
TiberiuTiberiu
161
New contributor
Tiberiu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
$begingroup$
Generally speaking you will run into troubles if you perform the fourier-transform with respect to the $E$ variable since the functions $f$ and $g$ depend on $E$ (this will lead to convolutions of your unknown function). If I were you I would try a laplace transform in $x$ and solve the resulting ODE, then transform back.
$endgroup$
– maxmilgram
yesterday
$begingroup$
For the 2. part: what type of BC do you impose at $x=0$?
$endgroup$
– maxmilgram
15 hours ago
$begingroup$
I would impose $varphi(E,0) = f(E) = delta(E)$. However, other conditions might also work. For example $varphi(E,0) = const$.
$endgroup$
– Tiberiu
15 hours ago
$begingroup$
Wait, so you have specific choices for the functions (or rather distributions) $f$and $g$?
$endgroup$
– maxmilgram
12 hours ago
$begingroup$
Yes. $f(E) = aE^k $ and $g(E) = fracdfdE = akE^k-1$. I realize now that in my previous reply I identified the condition $varphi(E,0)=f(E)$. This $f(E)$ is not the same as the one from the differential equation
$endgroup$
– Tiberiu
10 hours ago
add a comment |
2
2
2
$begingroup$
The question is really in the title. I have been seeing many examples of PDE's (heat equation on an infinite domain for example) being solved using Fourier transforms (FT). However, I have been unable to find a theory that says which type of PDE's can be solved by FT. So, my questions would be:
- What does mathematical theory say regarding the type of PDE's that can be solved by FT?
- Is the following PDE solvable by FT : $c cdot varphi_EE + f(E) cdot varphi_E + g(E) cdot varphi + varphi_x = 0 $, where $varphi = varphi(E,x)$, $E in (-infty,+infty)$, $ x in (0,infty)$, $varphi_EE=fracpartial^2varphipartial E^2$ and $varphi_x=fracpartialvarphipartial x$ and $c = constant$
- If the above equation is not solvable by Fourier Transforms, what other methods (aside from numerical solutions) are applicable?
Thank you in advance
pde fourier-transform
asked yesterday
TiberiuTiberiu
161
New contributor
Tiberiu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
The question is really in the title. I have been seeing many examples of PDE's (heat equation on an infinite domain for example) being solved using Fourier transforms (FT). However, I have been unable to find a theory that says which type of PDE's can be solved by FT. So, my questions would be:
- What does mathematical theory say regarding the type of PDE's that can be solved by FT?
- Is the following PDE solvable by FT : $c cdot varphi_EE + f(E) cdot varphi_E + g(E) cdot varphi + varphi_x = 0 $, where $varphi = varphi(E,x)$, $E in (-infty,+infty)$, $ x in (0,infty)$, $varphi_EE=fracpartial^2varphipartial E^2$ and $varphi_x=fracpartialvarphipartial x$ and $c = constant$
- If the above equation is not solvable by Fourier Transforms, what other methods (aside from numerical solutions) are applicable?
Thank you in advance
pde fourier-transform
pde fourier-transform
asked yesterday
TiberiuTiberiu
161
New contributor
Tiberiu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
TiberiuTiberiu
161
New contributor
Tiberiu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
TiberiuTiberiu
161
New contributor
Tiberiu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked yesterday
TiberiuTiberiu
161
asked yesterday
TiberiuTiberiu
161
161
New contributor
Tiberiu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Tiberiu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Tiberiu is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$begingroup$
Generally speaking you will run into troubles if you perform the fourier-transform with respect to the $E$ variable since the functions $f$ and $g$ depend on $E$ (this will lead to convolutions of your unknown function). If I were you I would try a laplace transform in $x$ and solve the resulting ODE, then transform back.
$endgroup$
– maxmilgram
yesterday
$begingroup$
For the 2. part: what type of BC do you impose at $x=0$?
$endgroup$
– maxmilgram
15 hours ago
$begingroup$
I would impose $varphi(E,0) = f(E) = delta(E)$. However, other conditions might also work. For example $varphi(E,0) = const$.
$endgroup$
– Tiberiu
15 hours ago
$begingroup$
Wait, so you have specific choices for the functions (or rather distributions) $f$and $g$?
$endgroup$
– maxmilgram
12 hours ago
$begingroup$
Yes. $f(E) = aE^k $ and $g(E) = fracdfdE = akE^k-1$. I realize now that in my previous reply I identified the condition $varphi(E,0)=f(E)$. This $f(E)$ is not the same as the one from the differential equation
$endgroup$
– Tiberiu
10 hours ago
add a comment |
$begingroup$
Generally speaking you will run into troubles if you perform the fourier-transform with respect to the $E$ variable since the functions $f$ and $g$ depend on $E$ (this will lead to convolutions of your unknown function). If I were you I would try a laplace transform in $x$ and solve the resulting ODE, then transform back.
$endgroup$
– maxmilgram
yesterday
$begingroup$
For the 2. part: what type of BC do you impose at $x=0$?
$endgroup$
– maxmilgram
15 hours ago
$begingroup$
I would impose $varphi(E,0) = f(E) = delta(E)$. However, other conditions might also work. For example $varphi(E,0) = const$.
$endgroup$
– Tiberiu
15 hours ago
$begingroup$
Wait, so you have specific choices for the functions (or rather distributions) $f$and $g$?
$endgroup$
– maxmilgram
12 hours ago
$begingroup$
Yes. $f(E) = aE^k $ and $g(E) = fracdfdE = akE^k-1$. I realize now that in my previous reply I identified the condition $varphi(E,0)=f(E)$. This $f(E)$ is not the same as the one from the differential equation
$endgroup$
– Tiberiu
10 hours ago
$begingroup$
Generally speaking you will run into troubles if you perform the fourier-transform with respect to the $E$ variable since the functions $f$ and $g$ depend on $E$ (this will lead to convolutions of your unknown function). If I were you I would try a laplace transform in $x$ and solve the resulting ODE, then transform back.
$endgroup$
– maxmilgram
yesterday
$begingroup$
Generally speaking you will run into troubles if you perform the fourier-transform with respect to the $E$ variable since the functions $f$ and $g$ depend on $E$ (this will lead to convolutions of your unknown function). If I were you I would try a laplace transform in $x$ and solve the resulting ODE, then transform back.
$endgroup$
– maxmilgram
yesterday
$begingroup$
For the 2. part: what type of BC do you impose at $x=0$?
$endgroup$
– maxmilgram
15 hours ago
$begingroup$
For the 2. part: what type of BC do you impose at $x=0$?
$endgroup$
– maxmilgram
15 hours ago
$begingroup$
I would impose $varphi(E,0) = f(E) = delta(E)$. However, other conditions might also work. For example $varphi(E,0) = const$.
$endgroup$
– Tiberiu
15 hours ago
$begingroup$
I would impose $varphi(E,0) = f(E) = delta(E)$. However, other conditions might also work. For example $varphi(E,0) = const$.
$endgroup$
– Tiberiu
15 hours ago
$begingroup$
Wait, so you have specific choices for the functions (or rather distributions) $f$and $g$?
$endgroup$
– maxmilgram
12 hours ago
$begingroup$
Wait, so you have specific choices for the functions (or rather distributions) $f$and $g$?
$endgroup$
– maxmilgram
12 hours ago
$begingroup$
Yes. $f(E) = aE^k $ and $g(E) = fracdfdE = akE^k-1$. I realize now that in my previous reply I identified the condition $varphi(E,0)=f(E)$. This $f(E)$ is not the same as the one from the differential equation
$endgroup$
– Tiberiu
10 hours ago
$begingroup$
Yes. $f(E) = aE^k $ and $g(E) = fracdfdE = akE^k-1$. I realize now that in my previous reply I identified the condition $varphi(E,0)=f(E)$. This $f(E)$ is not the same as the one from the differential equation
$endgroup$
– Tiberiu
10 hours ago
add a comment |
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$begingroup$
Generally speaking you will run into troubles if you perform the fourier-transform with respect to the $E$ variable since the functions $f$ and $g$ depend on $E$ (this will lead to convolutions of your unknown function). If I were you I would try a laplace transform in $x$ and solve the resulting ODE, then transform back.
$endgroup$
– maxmilgram
yesterday
$begingroup$
For the 2. part: what type of BC do you impose at $x=0$?
$endgroup$
– maxmilgram
15 hours ago
$begingroup$
I would impose $varphi(E,0) = f(E) = delta(E)$. However, other conditions might also work. For example $varphi(E,0) = const$.
$endgroup$
– Tiberiu
15 hours ago
$begingroup$
Wait, so you have specific choices for the functions (or rather distributions) $f$and $g$?
$endgroup$
– maxmilgram
12 hours ago
$begingroup$
Yes. $f(E) = aE^k $ and $g(E) = fracdfdE = akE^k-1$. I realize now that in my previous reply I identified the condition $varphi(E,0)=f(E)$. This $f(E)$ is not the same as the one from the differential equation
$endgroup$
– Tiberiu
10 hours ago