**Title:** Solving a Partial Differential Equation with Boundary Conditions**Objective:**This lesson focuses on solving a partial differential equation (PDE) given certain boundary conditions. The exercise considers specific boundary conditions and explores the solution in a region proximate to the bottom plate.**Problem Statement:**Consider the following partial differential equation (PDE) alongside its boundary conditions:\[\alpha y \frac{\partial C}{\partial x} = \mathcal{D} \frac{\partial^2 C}{\partial y^2}\]Boundary conditions are:\[C = C_0, \, y = 0\]\[C = C_1, \, y = L, \, x = 0\]**Context:**We analyze the scenario for short times when $ L $ is effectively infinitely distant from the bottom plate at $ y=0 $. We aim to solve the equation for the region near the bottom plate.**Tasks:**a) **Dimensionless Concentration ($ \theta $):**- Define a dimensionless concentration, $ \theta $, such that $ \theta = 0 $ at $ x = 0 $ and $ \theta = 1 $ at $ y = 0 $.- Write the corresponding equation and boundary conditions for $ \theta $.b) **Assume a Solution Form:**- Assume a solution of the form $ \theta = f(\eta) = f(y/g(x)) $.- Determine the function $ g(x) $.c) **Solve the Equation for $ \theta $:**- Solve the resulting equation for $ \theta $, which can be expressed as an integral (note that you are not required to integrate the integral in this step). This exercise will enhance understanding of dimensionless variables, boundary conditions, and the methodology for solving PDEs in specific applied contexts.

Consider the equationαγ∂C∂x=D∂2C∂y2C=c0,y=0 C=C1,y=L, x=0y=0 is a homogenous linear equation if δ≡0 it is a nonhomogenous linear equationαγ,D are called coffiecient of equation a A homogenous linear equation has a particular ω=0The principle of linear superposition holds ;namely if W1αγ w2D are particular solution to homogenous linear equations. +A2w2(x,y)+⋯+Anwn(x,y), where A1, A2, ..., An are arbitrary numbers is also an exact solution to that equation.

Math

Advanced Math

Consider the following partial differential equation and boundary conditions ay- D %D Ox C = C, y = 0 C =C,,y= L,x= 0 Consider the situation of short time when L is essentially infinitely far away from the bottom plate at y=0, i.e. solve the equation for the region close to the bottom plate. a) Define a dimensionless concentration, 0, such that 0=0 at x=0 and 0=1 at y=0. VWrite the equation and boundary conditions for 0

Consider the following partial differential equation and boundary conditions ay- D %D Ox C = C, y = 0 C =C,,y= L,x= 0 Consider the situation of short time when L is essentially infinitely far away from the bottom plate at y=0, i.e. solve the equation for the region close to the bottom plate. a) Define a dimensionless concentration, 0, such that 0=0 at x=0 and 0=1 at y=0. VWrite the equation and boundary conditions for 0

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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Concept explainers

Differential Equations

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differential equations.

Question

**Title:** Solving a Partial Differential Equation with Boundary Conditions

**Objective:**
This lesson focuses on solving a partial differential equation (PDE) given certain boundary conditions. The exercise considers specific boundary conditions and explores the solution in a region proximate to the bottom plate.

**Problem Statement:**
Consider the following partial differential equation (PDE) alongside its boundary conditions:

\[
\alpha y \frac{\partial C}{\partial x} = \mathcal{D} \frac{\partial^2 C}{\partial y^2}
\]

Boundary conditions are:

\[
C = C_0, \, y = 0
\]
\[
C = C_1, \, y = L, \, x = 0
\]

**Context:**
We analyze the scenario for short times when $ L $ is effectively infinitely distant from the bottom plate at $ y=0 $. We aim to solve the equation for the region near the bottom plate.

**Tasks:**

a) **Dimensionless Concentration ($ \theta $):**
- Define a dimensionless concentration, $ \theta $, such that $ \theta = 0 $ at $ x = 0 $ and $ \theta = 1 $ at $ y = 0 $.
- Write the corresponding equation and boundary conditions for $ \theta $.

b) **Assume a Solution Form:**
- Assume a solution of the form $ \theta = f(\eta) = f(y/g(x)) $.
- Determine the function $ g(x) $.

c) **Solve the Equation for $ \theta $:**
- Solve the resulting equation for $ \theta $, which can be expressed as an integral (note that you are not required to integrate the integral in this step).

This exercise will enhance understanding of dimensionless variables, boundary conditions, and the methodology for solving PDEs in specific applied contexts.

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