Fourier method/Fourier method

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Problem: Find the distribution of the concentration of the diffusing substance in an infinite layer \( 0 \leq x \leq \ell \), \( -\infty

11.5.2.1 Fourier method

6.15 $

Problem: The diffusing substance with concentration \( C_{0}= \) const is in an infinite layer \( -h \leq x \leq h,-\infty< \) \( y, z<+\infty \) and are kept there by impenetrable partitions, located at \( \pm h \) until the \( t=0 \) moment of time, when the partitions are removed and the diffusion process in a wider layer begins \( -H \leq x \leq \) \( H(H>h) \). Find the distribution of the concentration of the diffusing substance when \( t>0 \), if on the surfaces \( x= \pm H \) there is a mass exchange with the environment, having a constant concentration of the diffusing substance \( C_{1} \).

11.5.2.2 Fourier method

6.15 $

Problem: Find the temperature distribution \( u(r ; t) \) inside the ball \( r \leq l \), if the initial temperature distribution inside the ball \( \varphi(r) \) is given. \begin{tabular}{|c|c|} \hline\( \left.u\right|_{l} \) & \( \left.u\right|_{t=0}=\varphi \) \\ \hline\( \left.u\right|_{r=l}=0 \) & \( \left.u\right|_{t=0}=\frac{A}{l}\left(\frac{r^{2}}{12 l^{2}}-\frac{r}{4 l^{2}}\right) \) \\ \hline \end{tabular}

11.5.2.3 Fourier method

7.38 $

Problem: Solve the wave equation on the given segment, with the boundary conditions \( V(0, t)==V(l, t)=0 \) and given initial conditions, using the Fourier method: \[ \begin{array}{l} V_{t t}^{\prime \prime}=121 V_{x x}^{\prime \prime} \\ V(x, 0)=\left\{\begin{array}{l} \frac{x}{7}, \quad 0 \leq x \leq 14 \\ \frac{28-x}{7}, \quad 14 \leq x \leq 28 \end{array} \quad V_{t}^{\prime}(x, 0)=0 .\right. \end{array} \]

11.5.2.4 Fourier method

7.38 $

Problem: Solve the equation \( u_{x x}-\frac{1}{a^{2}} u_{t t}=0,(0 \leq x \leq l) \) under the given initial and boundary conditions. \begin{tabular}{|c|c|c|c|} \hline\( u(0, t) \) & \( u(l, t) \) & \( u(x, 0) \) & \( u_{t}^{\prime}(x, 0) \) \\ \hline 0 & 0 & \( 3 \sin \frac{5 \pi}{l} x \) & \( 7 x \) \\ \hline \end{tabular}

11.5.2.5 Fourier method

7.38 $

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}, \quad a=1, \quad u(0, x)=\frac{x}{2}, \\ \frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.6 Fourier method

3.2 $

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}, \quad a=2, \quad u(x, 0)=2 \cos \frac{5}{2} x, \\ \frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.7 Fourier method

3.2 $

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}, \quad a=3, \quad u(0, x)=x+2, \\ \frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.8 Fourier method

3.2 $

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}+t^{2} x^{2}, \\ a=3.5, u(x, 0)=\frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.9 Fourier method

3.69 $

Problem: Solve the boundary-value problem for the homogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}+t, \quad a=1.5 \\ u(x, 0)=\frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.10 Fourier method

3.69 $

Problem: Solve the boundary-value problem for the inhomogeneous wave equation: \[ \begin{array}{l} u_{t t}=a^{2} u_{x x}+t^{2} x, \quad a=2, \\ u(0, x)=\frac{\partial u}{\partial t}(0, x)=u(t, 0)=u(t, l)=0 . \end{array} \]

11.5.2.11 Fourier method

4.18 $

Problem: Find the solution of the Cauchy problem for the heat equation: \[ \begin{array}{l} u_{t}=4 u_{x x}, \\ u(x, 0)=15 \sin 3 \pi x, u(0, t)=0, u_{x}(4.5, t)=0 . \end{array} \]

11.5.2.12 Fourier method

4.92 $

Problem: Find the solution of the Cauchy problem for the heat equation: \[ \begin{array}{l} u_{t}=4 u_{x x}, \\ u(x, 0)=5 \sin 3 \pi x, \quad u(0, t)=u(6, t)=0 . \end{array} \]

11.5.2.13 Fourier method

4.18 $

Problem: Find the solution of the Cauchy problem for the heat equation: \[ \begin{array}{l} u_{t}=8 u_{x x}, \quad u(x, 0)=5 \cos 2 \pi x+6 \cos 3 \pi x, \\ u_{x}(0, t)=u_{x}(6, t)=0, \quad a=\sqrt{8}=2 \sqrt{2}, \quad l=6 . \end{array} \]

11.5.2.14 Fourier method

4.92 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{ll} u_{t t}=25 u_{x x}, & u(x, 0)=5 \sin 3 \pi x, \\ u_{t}(x, 0)=0, & u(0, t)=u(3, t)=0 . \end{array} \]

11.5.2.15 Fourier method

2.95 $

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