(a) Central difference numerical gradients can be used when solving differential equations using the finite difference method. The approach is important for numerically solving many engineering equations.< An example of an ordinary differential equation (ODE) which represents a boundary value problem, where C is known at x =1 and x = 3.5 is given below: < d²c dc 2 +4 +8C = 4 C(1) = 7, C(3.5) = 2,< dx² dx Using a step size of h=0.5, formulate an appropriate matrix equation (in form x = c) to solve the given boundary value equation. There will be 4 unknown values of C (i.e. C₁, C₂, C3, C4). Show your working.< Using a computational tool of your choice (Excel, Matlab etc) solve the system of equations and graph your results to show C against x.<
(a) Central difference numerical gradients can be used when solving differential equations using the finite difference method. The approach is important for numerically solving many engineering equations.< An example of an ordinary differential equation (ODE) which represents a boundary value problem, where C is known at x =1 and x = 3.5 is given below: < d²c dc 2 +4 +8C = 4 C(1) = 7, C(3.5) = 2,< dx² dx Using a step size of h=0.5, formulate an appropriate matrix equation (in form x = c) to solve the given boundary value equation. There will be 4 unknown values of C (i.e. C₁, C₂, C3, C4). Show your working.< Using a computational tool of your choice (Excel, Matlab etc) solve the system of equations and graph your results to show C against x.<
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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Transcribed Image Text:3. (a) Central difference numerical gradients can be used when solving differential equations
using the finite difference method. The approach is important for numerically solving many
engineering equations.<
↑
고
An example of an ordinary differential equation (ODE) which represents a boundary
value problem, where C is known at x =1 and x = 3.5 is given below: <
d²c
2- +4
dx²
dc
dx
+8C4 C(1) = 7, C(3.5) = 2,<
Using a step size of h=0.5, formulate an appropriate matrix equation (in form x = c) to
solve the given boundary value equation. There will be 4 unknown values of C (i.e.
C₁, C₂, C3, C4). Show your working.<
ܢܢ
Using a computational tool of your choice (Excel, Matlab etc) solve the system of
equations and graph your results to show C against x.<
(b) (i) Sketch and label on a graph the location of the basis vectors after the
transformation A:<
(iii) Show AA-¹ = I
A
=
-L₁₁
-4
1
(ii) Write down the inverse of A by drawing on your previous graph where the basis
vectors should land after the inverse transformation. <
(Where I is the identity matrix)
J
2 2
I I
N
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