Answered: Example. Show that the solution (if…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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$### Example #### Show that the solution (if exists) is unique Consider the following partial differential equation and boundary conditions: \[ \begin{cases} u_t = 5u_{xx} + \sin(x) \cos(2t), & 0 < x < \pi, t > 0 \\ u_x(0,t) = \sin(t), \quad u_x(\pi,t) = \cos(t), & t > 0 \\ u(x,0) = \cos(2x), & 0 < x < \pi \end{cases} \] **Explanation:** 1. **Equation:** $ u_t = 5u_{xx} + \sin(x)\cos(2t) $ - This is a second-order partial differential equation representing the evolution of a function $ u(x, t) $ over time $ t $ and space $ x $. - The term $ u_t $ denotes the partial derivative of $ u $ with respect to time. - The term $ u_{xx} $ denotes the second partial derivative of $ u $ with respect to space. - The term $ \sin(x)\cos(2t) $ serves as a non-homogeneous source term in the equation. 2. **Boundary Conditions:** - $ u_x(0,t) = \sin(t) $: This specifies the value of the spatial derivative of $ u $ at the left boundary $ x = 0 $ for any time $ t $. - $ u_x(\pi,t) = \cos(t) $: This specifies the value of the spatial derivative of $ u $ at the right boundary $ x = \pi $ for any time $ t $. 3. **Initial Condition:** - $ u(x,0) = \cos(2x) $: This provides the initial profile of $ u $ at time $ t = 0 $ across the spatial domain $ 0 < x < \pi $. The task is to show that a unique solution exists for this PDE given the boundary and initial conditions.$

Transcribed Image Text:### Example #### Show that the solution (if exists) is unique Consider the following partial differential equation and boundary conditions: \[ \begin{cases} u_t = 5u_{xx} + \sin(x) \cos(2t), & 0 < x < \pi, t > 0 \\ u_x(0,t) = \sin(t), \quad u_x(\pi,t) = \cos(t), & t > 0 \\ u(x,0) = \cos(2x), & 0 < x < \pi \end{cases} \] **Explanation:** 1. **Equation:** $ u_t = 5u_{xx} + \sin(x)\cos(2t) $ - This is a second-order partial differential equation representing the evolution of a function $ u(x, t) $ over time $ t $ and space $ x $. - The term $ u_t $ denotes the partial derivative of $ u $ with respect to time. - The term $ u_{xx} $ denotes the second partial derivative of $ u $ with respect to space. - The term $ \sin(x)\cos(2t) $ serves as a non-homogeneous source term in the equation. 2. **Boundary Conditions:** - $ u_x(0,t) = \sin(t) $: This specifies the value of the spatial derivative of $ u $ at the left boundary $ x = 0 $ for any time $ t $. - $ u_x(\pi,t) = \cos(t) $: This specifies the value of the spatial derivative of $ u $ at the right boundary $ x = \pi $ for any time $ t $. 3. **Initial Condition:** - $ u(x,0) = \cos(2x) $: This provides the initial profile of $ u $ at time $ t = 0 $ across the spatial domain $ 0 < x < \pi $. The task is to show that a unique solution exists for this PDE given the boundary and initial conditions.

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