Mixed problems

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Problem: Solve the mixed problem: \[ \begin{array}{l} u_{t t}=u_{x x}+5 u+t, 00, \\ \left.u\right|_{x=0}=0,\left.u\right|_{x=\pi}=0, \\ \left.u\right|_{t=0}=\sin 4 x,\left.u_{t}\right|_{t=0}=0 . \end{array} \]

11.5.5.1 Mixed problems

7.42 $

Problem: Solve the mixed problem: \[ \left\{\begin{array}{c} u_{t t}=u_{x x}+t^{2} x, \quad t>0 \\ \left.u\right|_{t=0}=x^{2},\left.\quad u_{t}\right|_{t=0}=0 \end{array}\right. \]

11.5.5.2 Mixed problems

2.97 $

Problem: Solve the mixed problem: \[ \left\{\begin{array}{c} u_{t t}=u_{x x}+6 t, \quad t>0, \quad x>0 \\ \left.u\right|_{t=0}=2 x,\left.\quad u_{t}\right|_{t=0}=0,\left.\quad u\right|_{\chi=0}=t^{3} \end{array}\right. \]

11.5.5.3 Mixed problems

3.71 $

Problem: Solve the mixed problem, using the Fourier method: \[ \left\{\begin{array}{l} u_{t}=u_{x x}, 00 \\ u(x, 0)=\cos ^{4} x-\sin ^{4} x \\ u(0, t)=u_{x}(\pi, t)=0 \end{array}\right. \]

11.5.5.4 Mixed problems

7.42 $

Problem: Solve the problem with stationary inhomogeneities using the Fourier method: \[ \begin{array}{ll} u_{t t}=u_{x x}+6 x, & u(x ; 0)=\sin ^{2} x-x^{3}, \\ u_{t}=(x ; 0)=0, & u(0 ; t)=0, \quad u(\pi ; t)=-\pi^{3} . \end{array} \]

11.5.5.5 Mixed problems

6.18 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=\frac{1}{36} u_{x x}+17 \cos 4 t \cdot \sin 6 x \\ u(x, 0)=5 \sin 18 x, \quad u(0, t)=u(\pi, t)=0 . \end{array} \]

11.5.5.6 Mixed problems

7.42 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=\frac{1}{25} u_{x x}+17 \sin 4 t \sin 5 x \\ u(x, 0)=5 \sin 15 x+3 \pi-2 x, u(0, t)=3 \pi, u(\pi, t)=\pi \end{array} \]

11.5.5.7 Mixed problems

7.42 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t t}=16 u_{x x}, u(0, t)=-4, u(1, t)=-1, \\ u(x, 0)=5 \sin 2 \pi x-4+3 x, \quad u_{t}(x, 0)=0 . \end{array} \]

11.5.5.8 Mixed problems

4.94 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t t}=\frac{1}{16} u_{x x}+50 e^{-7 t} \cdot \sin 4 x, \\ u(x, 0)=u_{t}(x, 0)=0, \quad u(0, t)=u(\pi, t)=0 . \end{array} \]

11.5.5.9 Mixed problems

6.18 $

Problem: Find the solution of the mixed problem: \[ \left\{\begin{array}{l} u_{t}=u_{x x}+u-x+2 \sin 2 x \cos x, \quad u=u(x, t) \\ 0

11.5.5.10 Mixed problems

9.89 $

Problem: Solve the problem of longitudinal oscillations of a rod suspended by the end \( \mathrm{x}=0 \) (the end \( x=l \) is free), performed under the influence of gravity.

11.5.5.11 Mixed problems

8.65 $

Problem: A thin homogeneous rod is given \( 0

11.5.5.12 Mixed problems

8.65 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=9 u_{x x}, \quad u(x, 0)=9 \sin 3 \pi x-5+2 x, \\ u(0, t)=-5, \quad u(3, t)=1 . \end{array} \]

11.5.5.13 Mixed problems

6.18 $

Problem: Find the solution of the mixed problem: \[ \begin{array}{l} u_{t}=\frac{1}{16} u_{x x}+10 \sin 3 t \cdot \sin 4 x, \quad u(x, 0)=0, \\ u(0, t)=0, \quad u(\pi, t)=0 . \end{array} \]

11.5.5.14 Mixed problems

6.18 $

Problem: Solve the external Dirichlet boundary-value problem for the circle: \[ \begin{array}{l} \Delta u(r, \varphi)=0, \quad r>r_{0}, \quad 0 \leq \varphi<2 \pi, \\ |u(r, \varphi)|<+\infty, \quad u\left(r_{0}, \varphi\right)=\frac{u_{0} \sin \varphi}{25+7 \cos \varphi} . \end{array} \]

11.5.5.15 Mixed problems

9.89 $