The function y(t) satisfies the ordinary differential equation dy = f(t, y) for t> 0. dt (4) The variable t takes discrete values, t with a constant step-size h = tn+1 − tn, for all n Є N. The approximation of y(tn) is denoted yn. (a) Derive an implicit scheme by integrating (4) over the interval [tn,tn+1], using the trapezium quadrature rule. [5] (b) Consider the case f(y) -Ay where > 0 is a constant. == (i) Find the exact solution of equation (4), subject to the initial condition y(0) = yo, and identify the behaviour of the solution as t→ +∞. [2] (ii) Write down the difference equation corresponding to the implicit scheme derived in part (a) and show that Yn+1 = 1-Xh/2 1+Xh/2 Yn. .[3] (iii) Solve the difference equation (i.e. write down y in terms of yo) and show that this implicit scheme is unconditionally stable. [5]
The function y(t) satisfies the ordinary differential equation dy = f(t, y) for t> 0. dt (4) The variable t takes discrete values, t with a constant step-size h = tn+1 − tn, for all n Є N. The approximation of y(tn) is denoted yn. (a) Derive an implicit scheme by integrating (4) over the interval [tn,tn+1], using the trapezium quadrature rule. [5] (b) Consider the case f(y) -Ay where > 0 is a constant. == (i) Find the exact solution of equation (4), subject to the initial condition y(0) = yo, and identify the behaviour of the solution as t→ +∞. [2] (ii) Write down the difference equation corresponding to the implicit scheme derived in part (a) and show that Yn+1 = 1-Xh/2 1+Xh/2 Yn. .[3] (iii) Solve the difference equation (i.e. write down y in terms of yo) and show that this implicit scheme is unconditionally stable. [5]
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![The function y(t) satisfies the ordinary differential equation
dy
= f(t, y) for t> 0.
dt
(4)
The variable t takes discrete values, t with a constant step-size h = tn+1 − tn, for all n Є N.
The approximation of y(tn) is denoted yn.
(a) Derive an implicit scheme by integrating (4) over the interval [tn,tn+1], using the
trapezium quadrature rule. [5]
(b) Consider the case f(y)
-Ay where > 0 is a constant.
==
(i) Find the exact solution of equation (4), subject to the initial condition y(0) = yo,
and identify the behaviour of the solution as t→ +∞. [2]
(ii) Write down the difference equation corresponding to the implicit scheme derived
in part (a) and show that
Yn+1 =
1-Xh/2
1+Xh/2
Yn. .[3]
(iii) Solve the difference equation (i.e. write down y in terms of yo) and show that
this implicit scheme is unconditionally stable. [5]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3011c556-643e-4a01-b0e0-55d8cf24eddf%2F792bf073-b989-41ea-82a2-7452d8ca2cc9%2F2y8s4eq_processed.png&w=3840&q=75)
Transcribed Image Text:The function y(t) satisfies the ordinary differential equation
dy
= f(t, y) for t> 0.
dt
(4)
The variable t takes discrete values, t with a constant step-size h = tn+1 − tn, for all n Є N.
The approximation of y(tn) is denoted yn.
(a) Derive an implicit scheme by integrating (4) over the interval [tn,tn+1], using the
trapezium quadrature rule. [5]
(b) Consider the case f(y)
-Ay where > 0 is a constant.
==
(i) Find the exact solution of equation (4), subject to the initial condition y(0) = yo,
and identify the behaviour of the solution as t→ +∞. [2]
(ii) Write down the difference equation corresponding to the implicit scheme derived
in part (a) and show that
Yn+1 =
1-Xh/2
1+Xh/2
Yn. .[3]
(iii) Solve the difference equation (i.e. write down y in terms of yo) and show that
this implicit scheme is unconditionally stable. [5]
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