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3. Neumann Boundary (a) Derive the equation for approximating the Neumann boundary of this system using a first order accurate approximation for the derivative. (b) How could you achieve a higher order of accuracy for this boundary? (c) Now, derive the equation for approximating the Neumann boundary of this system using the ghost node approach. (Note: remember your final answer should only include points in your grid!)

3. Neumann Boundary (a) Derive the equation for approximating the Neumann boundary of this system using a first order accurate approximation for the derivative. (b) How could you achieve a higher order of accuracy for this boundary? (c) Now, derive the equation for approximating the Neumann boundary of this system using the ghost node approach. (Note: remember your final answer should only include points in your grid!)

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter5: Inverse, Exponential, And Logarithmic Functions
Section: Chapter Questions
Problem 18T
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For all following problems, consider the differential equation for steady-state heat transfer in a square plate, V²T = m²(T – T); T(x,y) where 0<r< L, 0<y<L T(x,0) = 0, T(1, y) = 0, T(r, 1) = T,, (0, y) y? %3D 1. Interior Points (a) Using Central difference approximations for the derivatives derive the equation for approximating the interior points of this system. Write your final answer in the following form: a,T;-1j + a,T+1j +a3Tj+1+a,Tj-1 +a;T = b where a are constant coefficients and b includes any forcing terms. 2. Dirichlet Boundaries Assume Ar Ay = h = 1/4. (a) Fill in the blanks. This is a by grid, with points. (b) Find the equations for approximating the three Dirichlet boundary conditions. Label them bottom, right, top, or left as appropriate. 3. Neumann Boundary (a) Derive the equation for approximating the Neumann boundary of this system using a first order accurate approximation for the derivative. (b) How could you achieve a higher order of accuracy for this boundary? (c) Now, derive the equation for approximating the Neumann boundary of this system using the ghost node approach. (Note: remember your final answer should only include points in your grid!)
Transcribed Image Text:For all following problems, consider the differential equation for steady-state heat transfer in a square plate, V²T = m²(T – T); T(x,y) where 0<r< L, 0<y<L T(x,0) = 0, T(1, y) = 0, T(r, 1) = T,, (0, y) y? %3D 1. Interior Points (a) Using Central difference approximations for the derivatives derive the equation for approximating the interior points of this system. Write your final answer in the following form: a,T;-1j + a,T+1j +a3Tj+1+a,Tj-1 +a;T = b where a are constant coefficients and b includes any forcing terms. 2. Dirichlet Boundaries Assume Ar Ay = h = 1/4. (a) Fill in the blanks. This is a by grid, with points. (b) Find the equations for approximating the three Dirichlet boundary conditions. Label them bottom, right, top, or left as appropriate. 3. Neumann Boundary (a) Derive the equation for approximating the Neumann boundary of this system using a first order accurate approximation for the derivative. (b) How could you achieve a higher order of accuracy for this boundary? (c) Now, derive the equation for approximating the Neumann boundary of this system using the ghost node approach. (Note: remember your final answer should only include points in your grid!)
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