Let-u-1-u-2-s-t-u-tt-1-2-x-1-2-2-x-i-2-u-1-u-1-x-1-x-2-0-x-1-x-2-u-1-x-1-x-2-0-0-u-tt-2-2-x-1-2-

Question Number 215550 by MrGaster last updated on 10/Jan/25

$$\boldsymbol{\mathrm{Let}}\:\boldsymbol{{u}}^{\left(\mathrm{1}\right)} ,\boldsymbol{{u}}^{\left(\mathrm{2}\right)} \boldsymbol{\mathrm{s}}.\boldsymbol{\mathrm{t}}.\begin{cases}{\boldsymbol{{u}}_{\boldsymbol{{tt}}} ^{\left(\mathrm{1}\right)} =\left(\frac{\partial^{\mathrm{2}} }{\partial\boldsymbol{{x}}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} }{\partial\boldsymbol{{x}}_{{i}} ^{\mathrm{2}} }\right)\boldsymbol{{u}}^{\left(\mathrm{1}\right)} }\\{\boldsymbol{{u}}^{\left(\mathrm{1}\right)} \left(\boldsymbol{{x}}_{\mathrm{1}} ,\boldsymbol{{x}}_{\mathrm{2}} ,\mathrm{0}\right)=\boldsymbol{\psi}\left(\boldsymbol{{x}}_{\mathrm{1}} ,\boldsymbol{{x}}_{\mathrm{2}} \right)}\\{\boldsymbol{{u}}^{\left(\mathrm{1}\right)} \left(\boldsymbol{{x}}_{\mathrm{1}} ,\boldsymbol{{x}}_{\mathrm{2}} ,\mathrm{0}\right)=\mathrm{0}}\end{cases},\begin{cases}{\boldsymbol{{u}}_{\boldsymbol{{tt}}} ^{\left(\mathrm{2}\right)} =\left(\frac{\partial^{\mathrm{2}} }{\partial\boldsymbol{{x}}_{\mathrm{1}} ^{\mathrm{2}} }+\frac{\partial^{\mathrm{2}} }{\partial\boldsymbol{{x}}_{\mathrm{2}} ^{\mathrm{2}} }+\boldsymbol{{c}}^{\mathrm{2}} \right)\boldsymbol{{u}}^{\left(\mathrm{2}\right)} }\\{\boldsymbol{{u}}^{\left(\mathrm{2}\right)} \left(\boldsymbol{{x}}_{\mathrm{1}} \boldsymbol{{x}}_{\mathrm{2}} ,\mathrm{0}\right)=\mathrm{0}}\\{\boldsymbol{{u}}_{\boldsymbol{{t}}} ^{\left(\mathrm{2}\right)} \left(\boldsymbol{{x}}_{\mathrm{1}} ,\boldsymbol{{x}}_{\mathrm{2}} ,\mathrm{0}\right)=\boldsymbol{\psi}\left(\boldsymbol{{x}}_{\mathrm{1}} ,\boldsymbol{{x}}_{\mathrm{2}} \right)}\end{cases} \\ $$$$\mathrm{prove}:\boldsymbol{{u}}^{\left(\mathrm{2}\right)} \left(\boldsymbol{{x}}_{\mathrm{1}} ,\boldsymbol{{x}}_{\mathrm{2}} ,\boldsymbol{{t}}\right)=\frac{\mathrm{1}}{\mathrm{2}\boldsymbol{\pi}}\int\int_{\boldsymbol{\xi}_{\mathrm{1}} ^{\mathrm{2}} +\boldsymbol{\xi}_{\mathrm{2}} ^{\mathrm{2}} \leq\boldsymbol{{t}}^{\mathrm{2}} } \frac{\boldsymbol{{e}}^{\boldsymbol{\xi}_{\mathrm{2}} \boldsymbol{{c}}} \boldsymbol{{u}}^{\left(\mathrm{1}\right)} \left(\boldsymbol{{x}}_{\mathrm{1}} ,\boldsymbol{{x}}_{\mathrm{2}} ,\boldsymbol{\xi}_{\mathrm{1}} \right)\boldsymbol{{d}\xi}_{\mathrm{1}} \boldsymbol{{d}\xi}_{\mathrm{2}} }{\:\sqrt{\boldsymbol{{t}}^{\mathrm{2}} −\boldsymbol{\xi}_{\mathrm{1}} ^{\mathrm{2}} −\boldsymbol{\xi}_{\mathrm{2}} ^{\mathrm{2}} }} \\ $$

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