\[ \begin {array}{rclcl} u_{t} &=& 4u_{xx},&\qquad& t>0,\ x\in \mathbb {R} \\[0.8em] u(0,x) &=& \varphi (x),&& x\in \mathbb {R} \\[1.2em] &&&& \end {array} \] \[ \varphi (x)=\begin {cases}\frac {1}{2\pi }(1+\cos x), & |x|\le \pi \\ 0, & |x|>\pi \end {cases} \]
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\[ \begin {array}{rclcl} u_{t} &=& 4u_{xx},&\qquad& t>0,\ x>0 \\[0.8em] u(0,x) &=& \varphi (x),&& x>0 \\ u(t,0) &=& 0,&& t>0\\[1.2em] &&&& \end {array} \] \[ \varphi (x)=\begin {cases}\frac {1}{2\pi }\big (1+\cos (x-11)\big ), & |x-11|\le \pi \\ 0, & |x|>\pi \end {cases} \]
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\[ \begin {array}{rclcl} u_{t} &=& 4u_{xx},&\qquad& t>0,\ x>0 \\[0.8em] u(0,x) &=& \varphi (x),&& x>0 \\ u_x(t,0) &=& 0,&& t>0 \\[1.2em] &&&& \end {array} \] \[ \varphi (x)=\begin {cases}\frac {1}{2\pi }\big (1+\cos (x-11)\big ), & |x-11|\le \pi \\ 0, & |x|>\pi \end {cases} \]
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\[ \begin {array}{rclcl} u_{t} &=& 4u_{xx},&\qquad& t>0,\ 0<x<l \\[0.8em] u(0,x) &=& \varphi (x),&& 0<x<l \\ u(t,0) = u(t,l) &=& 0,&& t>0 \\[1.2em] &&&& \end {array} \] \[ \varphi (x)=\begin {cases}\frac {1}{2\pi }\big (1+\cos (x-11)\big ), & |x-11|\le \pi \\ 0, & |x|>\pi \end {cases} \]
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\[ \begin {array}{rclcl} u_{t} &=& 4u_{xx},&\qquad& t>0,\ x>0 \\[0.8em] u(t,0) &=& \frac {3}{10}\cos \frac 14t,&& t>0 \\[1.8em] &&&& \end {array} \]
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\[ \begin {array}{rclcl} u_{t} &=& 4u_{xx},&\qquad& t>0,\ 0<x<l \\[0.8em] u(0,x) &=& 0,&& 0<x<l \\ u(t,0)\ =\ 1,\ u(t,l) &=& 0, && t>0 \\[1.2em] &&&& \end {array} \]
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\[ \begin {array}{rclcl} u_{t} &=& 4u_{xx},&\qquad& t>0,\ 0<x<l \\[0.8em] u(0,x) &=& 0,&& 0<x<l \\ u(t,0)\ =\ 1,\ u_x(t,l) &=& 0,&& t>0 \\[1.2em] &&&& \end {array} \]
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\[ \begin {array}{rclcl} u_{t} &=& 4u_{xx},&\qquad& t>0,\ 0<x<l \\[0.8em] u(0,x) &=& 0,&& 0<x<l \\ u(t,0)\ =\ 1,\ u_x(t,l) &=& -u(t,l),&& t>0 \\[1.2em] &&&& \end {array} \]
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