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

11.5.2.28 Fourier method

7.49 $

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

6.24 $

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

6.24 $

Problem: Solve the Poisson equation \( u_{x x}+u_{y y}=1 \) in the ring \( r \leq \sqrt{x^{2}+y^{2}} \leq R \). The values of the function \( u(x, y) \) on the inner and outer boundaries of the ring are equal relatively to \( a, b \).

11.5.2.31 Fourier method

7.49 $

Problem: Find the solution of the Laplace equation in the circular sector \( 0

11.5.2.32 Fourier method

5.49 $

Problem: The Fourier method for the Laplace equation outside the circle. - Find the representation of the solution of the twodimensional Laplace equation \( u_{x x}+u_{y y}=0 \) in the given region, in the form of a series, performing separation of variables in polar coordinates; - sum the series, present the solution in the form of an integral; - calculate the solution for the boundary values, shown in the table. The region is outside the circle \( r \geq R \Rightarrow\left(r=\sqrt{x^{2}+y^{2}}\right) \Rightarrow \) \[ \Rightarrow\left\{\begin{array}{l} \Delta u=0, \quad r \geq R \\ \left.\left(u_{r}-u\right)\right|_{r=R}=\theta(\varphi) \end{array}\right. \] the boundary conditon is \( \theta(\varphi)=\sin 2 \varphi \).

11.5.2.33 Fourier method

8.74 $

Problem: Find the functions \( u(\rho, \varphi) \), that are harmonic outside the circle of radius \( \rho=a \) and satisfying the boundary condition (first exterior boundary value problem for a circle): \[ \left.u\right|_{\rho=a}=A \text {. } \]

11.5.2.34 Fourier method

6.24 $

Problem: Find the solution of the exterior boundary value problem for the Laplace equation, if on the boundary of the circle the conditions are given: \[ \left.u\right|_{\rho-a}=A \sin ^{3} \varphi+B \text {. } \]

11.5.2.35 Fourier method

8.74 $

Problem: Find the functions \( u(r, \varphi) \), harmonic outside the circle radius \( r=a \) and satisfying the boundary condition (the first exterior boundary-value for the circle): \[ \left.\frac{\partial u}{\partial r}\right|_{r=a}=A \cos 3 \varphi \text {. } \]

11.5.2.36 Fourier method

7.49 $

Problem: Determine the type of the differential equation, bring it to the canonical form, write down the general solution, find the solution of the Cauchy problem. \[ \begin{array}{l} y^{4} u_{x x}+2 y^{2} u_{x y}+u_{y y}-\frac{2}{y} u_{y}=0, \\ u(x, 1)=\frac{x^{3}}{3}, \quad u_{y}(x, 1)=2 x . \end{array} \]

11.5.3.6 With variable coefficients

7.49 $

Problem: Bring the following equation to the canonical form in each of the regions, where the type of the considered equation is preserved: \[ x^{2} u_{x x}+2 x y u_{x y}+y^{2} u_{y y}-2 y u_{x}+y e^{\frac{y}{x}}=0 . \]

11.5.3.7 With variable coefficients

5.49 $

Problem: Find the general solution of the equation, bringing it to canonical form: \[ u_{x x}+12 u_{x y}+36 u_{y y}+u_{x}+6 u_{y}=0 . \]

11.5.4.13 With constant coefficients

2.5 $

Problem: Determine the type of the second order equation. \[ 4 u_{x x}-4 u_{x y}-2 u_{y z}+u_{y}+u_{z}=0 . \]

11.5.4.14 With constant coefficients

1.25 $

Problem: Bring the second order linear partial differential equation to canonical form. \[ \frac{\theta^{2} u}{\theta x^{2}}-2 \frac{\theta^{2} u}{\theta x \theta y}+\frac{\theta^{2} u}{\theta y^{2}}+\frac{\theta u}{\theta x}+\frac{\theta u}{\theta y}+u=0 \]

11.5.4.15 With constant coefficients

2.5 $

Problem: Bring the second order linear partial differential equation to canonical form. \[ \frac{\theta^{2} u}{\theta x^{2}}+2 \frac{\theta^{2} u}{\theta x \theta y}+4 \frac{\theta^{2} u}{\theta y^{2}}=-2 \frac{\theta u}{\theta x}-3 \frac{\theta u}{\theta y} . \]

11.5.4.16 With constant coefficients

2.5 $

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