Can this integral (of PDE solution) be solved?
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I study the heat equation from the book, Heat Conduction, Third Edition.
By using the method of separation of variables for the boundary condition of
$$BC1: frac{partial T}{partial x} Bigg|_{x=0}= 0$$
$$IC: T(x,t=0)=F(x)$$
the following solution is given,
$$T(x,t) = frac{1}{(4 pi alpha t)^{1/2}} int_{x'=0}^{infty} F(x')Bigglbraceexp left [ -frac{(x-x')^2}{4 alpha t}right ] + exp left [ -frac{(x+x')^2}{4 alpha t}right ] Biggrbrace dx'$$
Surprisingly, the section ends with this equation. I wonder if this integration can be generally solved to obtain a more tangible solution for the temperature.
calculus integration pde
asked Jan 25 at 20:57
KimiaKimia
314
$endgroup$
add a comment |
$begingroup$
I study the heat equation from the book, Heat Conduction, Third Edition.
By using the method of separation of variables for the boundary condition of
$$BC1: frac{partial T}{partial x} Bigg|_{x=0}= 0$$
$$IC: T(x,t=0)=F(x)$$
the following solution is given,
$$T(x,t) = frac{1}{(4 pi alpha t)^{1/2}} int_{x'=0}^{infty} F(x')Bigglbraceexp left [ -frac{(x-x')^2}{4 alpha t}right ] + exp left [ -frac{(x+x')^2}{4 alpha t}right ] Biggrbrace dx'$$
Surprisingly, the section ends with this equation. I wonder if this integration can be generally solved to obtain a more tangible solution for the temperature.
calculus integration pde
asked Jan 25 at 20:57
KimiaKimia
314
$endgroup$
2
$begingroup$
In general, no. Even more so, you don’t gain anything from the evaluates form that you can’t see from the integral solution.
$endgroup$
– DaveNine
Jan 26 at 0:10
add a comment |
$begingroup$
I study the heat equation from the book, Heat Conduction, Third Edition.
By using the method of separation of variables for the boundary condition of
$$BC1: frac{partial T}{partial x} Bigg|_{x=0}= 0$$
$$IC: T(x,t=0)=F(x)$$
the following solution is given,
$$T(x,t) = frac{1}{(4 pi alpha t)^{1/2}} int_{x'=0}^{infty} F(x')Bigglbraceexp left [ -frac{(x-x')^2}{4 alpha t}right ] + exp left [ -frac{(x+x')^2}{4 alpha t}right ] Biggrbrace dx'$$
Surprisingly, the section ends with this equation. I wonder if this integration can be generally solved to obtain a more tangible solution for the temperature.
calculus integration pde
asked Jan 25 at 20:57
KimiaKimia
314
$endgroup$
I study the heat equation from the book, Heat Conduction, Third Edition.
By using the method of separation of variables for the boundary condition of
$$BC1: frac{partial T}{partial x} Bigg|_{x=0}= 0$$
$$IC: T(x,t=0)=F(x)$$
the following solution is given,
$$T(x,t) = frac{1}{(4 pi alpha t)^{1/2}} int_{x'=0}^{infty} F(x')Bigglbraceexp left [ -frac{(x-x')^2}{4 alpha t}right ] + exp left [ -frac{(x+x')^2}{4 alpha t}right ] Biggrbrace dx'$$
Surprisingly, the section ends with this equation. I wonder if this integration can be generally solved to obtain a more tangible solution for the temperature.
calculus integration pde
calculus integration pde
asked Jan 25 at 20:57
KimiaKimia
314
asked Jan 25 at 20:57
KimiaKimia
314
asked Jan 25 at 20:57
KimiaKimia
314
asked Jan 25 at 20:57
KimiaKimia
314
asked Jan 25 at 20:57
KimiaKimia
314
314
2
$begingroup$
In general, no. Even more so, you don’t gain anything from the evaluates form that you can’t see from the integral solution.
$endgroup$
– DaveNine
Jan 26 at 0:10
add a comment |
2
$begingroup$
In general, no. Even more so, you don’t gain anything from the evaluates form that you can’t see from the integral solution.
$endgroup$
– DaveNine
Jan 26 at 0:10
2
2
$begingroup$
In general, no. Even more so, you don’t gain anything from the evaluates form that you can’t see from the integral solution.
$endgroup$
– DaveNine
Jan 26 at 0:10
$begingroup$
In general, no. Even more so, you don’t gain anything from the evaluates form that you can’t see from the integral solution.
$endgroup$
– DaveNine
Jan 26 at 0:10
add a comment |
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$begingroup$
In general, no. Even more so, you don’t gain anything from the evaluates form that you can’t see from the integral solution.
$endgroup$
– DaveNine
Jan 26 at 0:10