Biomedical Engineering Reference
In-Depth Information
∂
2
u
∂x∂y
u
i
+
1,
j
+
1
−
u
i
+
1,
j
−
1
−
u
i
−
1,
j
+
1
+
u
i
−
1,
j
−
1
i
,
j
=
4
xy
i-
1
,j+
1
i,j+
1
i+
1
,j+
1
Δ
y
i-
1
,j
i,j
i+
1
,j
Δ
y
i-
1
,j-
1
i,j-
1
i+
1
,j-
1
Δ
x
Δ
x
30. Explain the difference between implicit and explicit methods in solving Ordinary
Differential Equations (ODEs).
31. Explain the difference in obtaining the gradient of a curve using the forward
Euler, Runge-Kutta, and the backward Euler methods.
32. Solve the following ODE
∂
∂t
=−
u
, using the explicit forward Euler method
=
=
where
u
(0)
0
.
2s.
Compare the errors between the numerical method with the analytical solution
which is given as,
u
1 using
h
1
exp (
t
)
.
Repeating the solution but using
h
=
=
0
.
4 s, and 0.1 s, determine the new errors.
33. Solve the following ODE
∂
∂t
=
sin (
u
), using the explicit forward Euler,
Runge-Kutta, and implicit backward Euler methods.
u
(0)
0
.
2s.
Compare the errors between the numerical methods with the analytical solution
which is given as,
u
=
1 using
h
=
2 tan
−
1
( exp (
t
=
+
ln ( tan (1)))
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