Proof: $F(x,t) := t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t})$ is solution of $Delta F - frac{partial F}{partial...












1














How can one prove that the function



$F: mathbb{R}^n times mathbb{R}^+ text{ {0}} to mathbb{R}$ with



$$F(x,t) := t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t})$$



is a solution of the partial differential equation



$$Delta F - frac{partial F}{partial t} = 0$$



I know that this can be done using the Laplace operator, which is given by $Delta := sum_{i=1}^n frac{partial^2}{partial x_i^2}$, but I don't know what the partial derivative of $Delta F - frac{partial F}{partial t} = 0$ looks like.






analysis functions partial-derivative







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  • Try with $n=2$, $F(x_1,x_2,t) = t^{-1} exp(-frac{x_1^2+x_2^2}{4t})$. What are $partial_{x_i}F$ and $partial_{x_i}^2F$ and $partial_t F$ and $Delta F$ ?
    – reuns
    Nov 23 '18 at 1:33


















1














How can one prove that the function



$F: mathbb{R}^n times mathbb{R}^+ text{ {0}} to mathbb{R}$ with



$$F(x,t) := t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t})$$



is a solution of the partial differential equation



$$Delta F - frac{partial F}{partial t} = 0$$



I know that this can be done using the Laplace operator, which is given by $Delta := sum_{i=1}^n frac{partial^2}{partial x_i^2}$, but I don't know what the partial derivative of $Delta F - frac{partial F}{partial t} = 0$ looks like.






analysis functions partial-derivative







share|cite|improve this question






















  • Try with $n=2$, $F(x_1,x_2,t) = t^{-1} exp(-frac{x_1^2+x_2^2}{4t})$. What are $partial_{x_i}F$ and $partial_{x_i}^2F$ and $partial_t F$ and $Delta F$ ?
    – reuns
    Nov 23 '18 at 1:33
















1












1








1







How can one prove that the function



$F: mathbb{R}^n times mathbb{R}^+ text{ {0}} to mathbb{R}$ with



$$F(x,t) := t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t})$$



is a solution of the partial differential equation



$$Delta F - frac{partial F}{partial t} = 0$$



I know that this can be done using the Laplace operator, which is given by $Delta := sum_{i=1}^n frac{partial^2}{partial x_i^2}$, but I don't know what the partial derivative of $Delta F - frac{partial F}{partial t} = 0$ looks like.






analysis functions partial-derivative







share|cite|improve this question













How can one prove that the function



$F: mathbb{R}^n times mathbb{R}^+ text{ {0}} to mathbb{R}$ with



$$F(x,t) := t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t})$$



is a solution of the partial differential equation



$$Delta F - frac{partial F}{partial t} = 0$$



I know that this can be done using the Laplace operator, which is given by $Delta := sum_{i=1}^n frac{partial^2}{partial x_i^2}$, but I don't know what the partial derivative of $Delta F - frac{partial F}{partial t} = 0$ looks like.






analysis functions partial-derivative




analysis functions partial-derivative






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Nov 22 '18 at 23:09









Bad At MathBad At Math

203




203












  • Try with $n=2$, $F(x_1,x_2,t) = t^{-1} exp(-frac{x_1^2+x_2^2}{4t})$. What are $partial_{x_i}F$ and $partial_{x_i}^2F$ and $partial_t F$ and $Delta F$ ?
    – reuns
    Nov 23 '18 at 1:33




















  • Try with $n=2$, $F(x_1,x_2,t) = t^{-1} exp(-frac{x_1^2+x_2^2}{4t})$. What are $partial_{x_i}F$ and $partial_{x_i}^2F$ and $partial_t F$ and $Delta F$ ?
    – reuns
    Nov 23 '18 at 1:33


















Try with $n=2$, $F(x_1,x_2,t) = t^{-1} exp(-frac{x_1^2+x_2^2}{4t})$. What are $partial_{x_i}F$ and $partial_{x_i}^2F$ and $partial_t F$ and $Delta F$ ?
– reuns
Nov 23 '18 at 1:33






Try with $n=2$, $F(x_1,x_2,t) = t^{-1} exp(-frac{x_1^2+x_2^2}{4t})$. What are $partial_{x_i}F$ and $partial_{x_i}^2F$ and $partial_t F$ and $Delta F$ ?
– reuns
Nov 23 '18 at 1:33












1 Answer
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It's very long, so let me help you with some expressions



$$Vert x Vert _2^{2} = x_1^2 + x_2^2 + cdots + x_n^2, $$



$$ dfrac{partial}{partial x_i}(x_1^2 + x_2^2 + cdots + x_n^2) = 2x_i, $$



$$dfrac{partial}{partial x_i}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i, $$



$$dfrac{partial^2}{partial x_i^2}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = dfrac{-1}{4t} Bigg[ exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i cdot 2 x_i + exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot 2Bigg] $$



$$dfrac{partial}{partial t} Big( t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t}) Big) = dfrac{-n t^{n/2-1}}{2} exp !Big(-dfrac{Vert x Vert_2^2}{4t} Big) + t^{-n/2} exp! Big(-frac{Vert x Vert_2^2}{4t} Big) cdot dfrac{Vert x Vert_2^2}{4t^2}.$$



Can you continue with the calculations?






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    active

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    It's very long, so let me help you with some expressions



    $$Vert x Vert _2^{2} = x_1^2 + x_2^2 + cdots + x_n^2, $$



    $$ dfrac{partial}{partial x_i}(x_1^2 + x_2^2 + cdots + x_n^2) = 2x_i, $$



    $$dfrac{partial}{partial x_i}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i, $$



    $$dfrac{partial^2}{partial x_i^2}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = dfrac{-1}{4t} Bigg[ exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i cdot 2 x_i + exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot 2Bigg] $$



    $$dfrac{partial}{partial t} Big( t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t}) Big) = dfrac{-n t^{n/2-1}}{2} exp !Big(-dfrac{Vert x Vert_2^2}{4t} Big) + t^{-n/2} exp! Big(-frac{Vert x Vert_2^2}{4t} Big) cdot dfrac{Vert x Vert_2^2}{4t^2}.$$



    Can you continue with the calculations?






    share|cite|improve this answer




























      1














      It's very long, so let me help you with some expressions



      $$Vert x Vert _2^{2} = x_1^2 + x_2^2 + cdots + x_n^2, $$



      $$ dfrac{partial}{partial x_i}(x_1^2 + x_2^2 + cdots + x_n^2) = 2x_i, $$



      $$dfrac{partial}{partial x_i}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i, $$



      $$dfrac{partial^2}{partial x_i^2}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = dfrac{-1}{4t} Bigg[ exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i cdot 2 x_i + exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot 2Bigg] $$



      $$dfrac{partial}{partial t} Big( t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t}) Big) = dfrac{-n t^{n/2-1}}{2} exp !Big(-dfrac{Vert x Vert_2^2}{4t} Big) + t^{-n/2} exp! Big(-frac{Vert x Vert_2^2}{4t} Big) cdot dfrac{Vert x Vert_2^2}{4t^2}.$$



      Can you continue with the calculations?






      share|cite|improve this answer


























        1












        1








        1






        It's very long, so let me help you with some expressions



        $$Vert x Vert _2^{2} = x_1^2 + x_2^2 + cdots + x_n^2, $$



        $$ dfrac{partial}{partial x_i}(x_1^2 + x_2^2 + cdots + x_n^2) = 2x_i, $$



        $$dfrac{partial}{partial x_i}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i, $$



        $$dfrac{partial^2}{partial x_i^2}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = dfrac{-1}{4t} Bigg[ exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i cdot 2 x_i + exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot 2Bigg] $$



        $$dfrac{partial}{partial t} Big( t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t}) Big) = dfrac{-n t^{n/2-1}}{2} exp !Big(-dfrac{Vert x Vert_2^2}{4t} Big) + t^{-n/2} exp! Big(-frac{Vert x Vert_2^2}{4t} Big) cdot dfrac{Vert x Vert_2^2}{4t^2}.$$



        Can you continue with the calculations?






        share|cite|improve this answer














        It's very long, so let me help you with some expressions



        $$Vert x Vert _2^{2} = x_1^2 + x_2^2 + cdots + x_n^2, $$



        $$ dfrac{partial}{partial x_i}(x_1^2 + x_2^2 + cdots + x_n^2) = 2x_i, $$



        $$dfrac{partial}{partial x_i}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i, $$



        $$dfrac{partial^2}{partial x_i^2}, exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) = dfrac{-1}{4t} Bigg[ exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot dfrac{-1}{4t} cdot 2x_i cdot 2 x_i + exp! Big(-dfrac{Vert x Vert_2^2}{4t} Big) cdot 2Bigg] $$



        $$dfrac{partial}{partial t} Big( t^{-n/2} exp(-frac{Vert x Vert_2^2}{4t}) Big) = dfrac{-n t^{n/2-1}}{2} exp !Big(-dfrac{Vert x Vert_2^2}{4t} Big) + t^{-n/2} exp! Big(-frac{Vert x Vert_2^2}{4t} Big) cdot dfrac{Vert x Vert_2^2}{4t^2}.$$



        Can you continue with the calculations?







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Nov 23 '18 at 2:23

























        answered Nov 23 '18 at 2:11









        DavidDavid

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