Fourier method

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Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} u_{t t}=25 u_{x x}, \quad u(0, t)=u(3, t)=0 \\ u(x, 0)=5 \sin 3 \pi x, \quad u_{t}(x, 0)=20 \pi \sin 4 \pi x . \end{array} \]

11.5.2.16 Fourier method

3.85 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{ll} u_{t t}=16 u_{x x}, & u_{x}(0, t)=u_{x}(4, t)=0, \\ u(x, 0)=0, & u_{t}(x, 0)=12 \pi \cos 3 \pi x . \end{array} \]

11.5.2.17 Fourier method

4.37 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} u_{t t}=9 u_{x x}, \quad u(x, 0)=5 \sin 5 \pi x, \\ u_{t}(x, 0)=0, \quad u(0, t)=0, \quad u_{x}(2.5, t)=0 . \end{array} \]

11.5.2.18 Fourier method

5.14 $

Problem: Solve the problem of oscillations of the string \( 0< \) \( x

11.5.2.19 Fourier method

6.42 $

Problem: Find the temperature distribution in the rod \( 0

11.5.2.20 Fourier method

7.71 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} u_{t t}=5 u_{x x}, \quad u(x, 0)=e^{-(x-1)^{2}}, \\ u_{t}(x, 0)=3 \cdot e^{-(x-1)^{2}}, \quad u(0, t)=u(2, t)=0 . \end{array} \]

11.5.2.21 Fourier method

4.37 $

Problem: Find the solution of the Cauchy problem for the string oscillation equation: \[ \begin{array}{l} V_{t t}^{\prime \prime}=9 V_{x x}^{\prime \prime}, \quad V(x, 0)=\left\{\begin{array}{l} \frac{3 x}{40}, \quad 0 \leq x \leq 4 \\ \frac{3(8-x)}{40}, \quad 4 \leq x \leq 8 \end{array},\right. \\ V(0, t)=V(8, t)=0, \quad t \in(0,+\infty) . \end{array} \]

11.5.2.22 Fourier method

7.71 $

Problem: Solve the problem of string oscillations, fixed at the ends, by the method of separation of variables \( x \) (the Fourier method). \[ \begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}}=16 \frac{\partial^{2} u}{\partial x^{2}}, \quad u(0, t)=u(3, t)=0 \\ u(x, 0)=\frac{8}{9}\left(3 x-x^{2}\right),\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=0 \end{array} \]

11.5.2.23 Fourier method

6.42 $

Problem: Solve the problem of string oscillations, fixed at the ends, by the method of separation of variables (the Fourier method). \[ \begin{array}{l} \frac{\partial^{2} u}{\partial t^{2}}=25 \frac{\partial^{2} u}{\partial x^{2}}, \quad u(0, t)=u(8, t)=0, \\ u(x, 0)=0,\left.\quad \frac{\partial u}{\partial t}\right|_{t=0}=\left\{\begin{array}{ll} x, & 0 \leq x<4 \\ 8-x, & 4 \leq x \leq 8 \end{array}\right. \end{array} \]

11.5.2.24 Fourier method

6.42 $

Problem: Find the solution of the homogeneous equation with the given initial and boundary conditions:

11.5.2.25 Fourier method

6.42 $

Problem: Find the solution of the Cauchy problem for the heatconduction equation: \[ \begin{array}{l} \frac{\partial u}{\partial t}-a^{2} \frac{\partial^{2} u}{\partial x^{2}}=0,0 \leq x \leq 1, u(0, t)=u(1, t)=0, \\ u(x, 0)=f(x)=\left\{\begin{array}{ll} x, & 0 \leq x \leq 0.5 \\ 1-x, & 0.5 \leq x \leq 1 \end{array}\right. \end{array} \]

11.5.2.26 Fourier method

5.91 $

Problem: Find the solution of the Cauchy problem for the heatconduction equation: \[ \left\{\begin{array}{l} \frac{\partial T}{\partial t}=a^{2} \frac{\partial^{2} T}{\partial x^{2}}, \quad(0 \leq x \leq 4) \\ \left.T\right|_{x=0}=T(0, t)=0 \\ \left.T\right|_{x=4}=T(4, t)=0 \\ T(x, 0)=f(x)=\left\{\begin{array}{ll} 0, & 0 \leq x<1 \\ 1, & 1 \leq x \leq 4 \end{array}\right. \end{array}\right. \]

11.5.2.27 Fourier method

6.42 $

Problem: Solve the Laplace equation, using the Fourier method:

11.5.2.28 Fourier method

7.71 $

Problem: Solve the Dirichlet problem for the Laplace equation in the circle: \( \Delta u=0 \) in the circle \( 0 \leq r<1,0 \leq \varphi<2 \pi \), \[ u(1, \varphi)=5 \cos 5 \varphi \text {. } \]

11.5.2.29 Fourier method

6.42 $

Problem: Solve the Laplace equation, using the Fourier method: \[ \left\{\begin{array}{l} \frac{\partial^{2} T}{\partial x^{2}}+\frac{\partial^{2} T}{\partial y^{2}}=0, \quad\left\{\begin{array}{l} 0 \leq x \leq 1 \\ 0 \leq y \leq 3 \end{array}\right. \\ T(0, y)=T(1, y)=0 \\ T(x, 0)=T(x, 3)=x-1 \end{array}\right. \]

11.5.2.30 Fourier method

6.42 $

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