IntroPDE Fundamental solution of the heat equation
運算式 9:
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Solution to the Cauchy Problem
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The initial condition
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運算式 12: "h" left parenthesis, "x" , right parenthesis equals natural log left parenthesis, StartAbsoluteValue, "x" , EndAbsoluteValue , right parenthesishx=lnx
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運算式 13: left parenthesis, "t" , 0 , "h" left parenthesis, "t" , right parenthesis , right parenthesist,0,ht
定義域 t 最小值： negative 5−5
less than or equal to "t" less than or equal to≤t≤
定義域 t 最大值： 55
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The formula for the solution from Theorem 4.7
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運算式 15: "u" left parenthesis, "x" , "t" , right parenthesis equals Start integral from negative 10 to 10, end integral, "F" left parenthesis, "x" minus "y" , "t" , right parenthesis "h" left parenthesis, "y" , right parenthesis "d" "y"ux,t=∫10−10Fx−y,thydy
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運算式 16: "z" equals "u" left parenthesis, "x" , "y" , right parenthesis left brace, "y" greater than or equal to 0 , right bracez=ux,yy≥0
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運算式 17: left parenthesis, "t" , "t" Subscript, 0 , Baseline , "u" left parenthesis, "t" , "t" Subscript, 0 , Baseline , right parenthesis , right parenthesist,t0,ut,t0
定義域 t 最小值： negative 5−5
less than or equal to "t" less than or equal to≤t≤
