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Solving heat equation by Duhamel's theorem
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The general solution for heat equation
$$u_t = - frac1alpha u_xx$$
with Duhamel's theorem yields to
$$u(x,t) = fracxsqrt4alpha t int_tau=0^t fracf(tau)(t-tau)^3/2expBig( fracx^24 alpha (t-tau)Big) dtau$$
How can I go further by applying the actual boundary condition of
$$f(t) = k e^-t$$
I assumed that the steps of solution to a famous PDE is known and I avoided, but if it is needed, I can post the whole solution.
pde heat-equation
edited Jul 31 at 18:24
Bernard
110k635102
asked Jul 31 at 17:47
Omani
61
add a comment |Â
up vote
1
down vote
favorite
The general solution for heat equation
$$u_t = - frac1alpha u_xx$$
with Duhamel's theorem yields to
$$u(x,t) = fracxsqrt4alpha t int_tau=0^t fracf(tau)(t-tau)^3/2expBig( fracx^24 alpha (t-tau)Big) dtau$$
How can I go further by applying the actual boundary condition of
$$f(t) = k e^-t$$
I assumed that the steps of solution to a famous PDE is known and I avoided, but if it is needed, I can post the whole solution.
pde heat-equation
edited Jul 31 at 18:24
Bernard
110k635102
asked Jul 31 at 17:47
Omani
61
You should simply put the initial condition in the integral and than solving it, doesn't seem like a pretty integral
– Davide Morgante
Jul 31 at 17:59
Are you sure that the solution if of that type?
– Davide Morgante
Jul 31 at 18:13
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
The general solution for heat equation
$$u_t = - frac1alpha u_xx$$
with Duhamel's theorem yields to
$$u(x,t) = fracxsqrt4alpha t int_tau=0^t fracf(tau)(t-tau)^3/2expBig( fracx^24 alpha (t-tau)Big) dtau$$
How can I go further by applying the actual boundary condition of
$$f(t) = k e^-t$$
I assumed that the steps of solution to a famous PDE is known and I avoided, but if it is needed, I can post the whole solution.
pde heat-equation
edited Jul 31 at 18:24
Bernard
110k635102
asked Jul 31 at 17:47
Omani
61
The general solution for heat equation
$$u_t = - frac1alpha u_xx$$
with Duhamel's theorem yields to
$$u(x,t) = fracxsqrt4alpha t int_tau=0^t fracf(tau)(t-tau)^3/2expBig( fracx^24 alpha (t-tau)Big) dtau$$
How can I go further by applying the actual boundary condition of
$$f(t) = k e^-t$$
I assumed that the steps of solution to a famous PDE is known and I avoided, but if it is needed, I can post the whole solution.
pde heat-equation
edited Jul 31 at 18:24
Bernard
110k635102
asked Jul 31 at 17:47
Omani
61
edited Jul 31 at 18:24
Bernard
110k635102
edited Jul 31 at 18:24
Bernard
110k635102
edited Jul 31 at 18:24
Bernard
110k635102
110k635102
asked Jul 31 at 17:47
Omani
61
asked Jul 31 at 17:47
Omani
61
asked Jul 31 at 17:47
Omani
61
61
You should simply put the initial condition in the integral and than solving it, doesn't seem like a pretty integral
– Davide Morgante
Jul 31 at 17:59
Are you sure that the solution if of that type?
– Davide Morgante
Jul 31 at 18:13
add a comment |Â
You should simply put the initial condition in the integral and than solving it, doesn't seem like a pretty integral
– Davide Morgante
Jul 31 at 17:59
Are you sure that the solution if of that type?
– Davide Morgante
Jul 31 at 18:13
You should simply put the initial condition in the integral and than solving it, doesn't seem like a pretty integral
– Davide Morgante
Jul 31 at 17:59
You should simply put the initial condition in the integral and than solving it, doesn't seem like a pretty integral
– Davide Morgante
Jul 31 at 17:59
Are you sure that the solution if of that type?
– Davide Morgante
Jul 31 at 18:13
Are you sure that the solution if of that type?
– Davide Morgante
Jul 31 at 18:13
add a comment |Â
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You should simply put the initial condition in the integral and than solving it, doesn't seem like a pretty integral
– Davide Morgante
Jul 31 at 17:59
Are you sure that the solution if of that type?
– Davide Morgante
Jul 31 at 18:13