Page 227 - Engineering Mathematics Workbook_Final

Numerical Methods

101. Numerical integration using (a) h (b) h
3
2
trapezoidal rule gives the best result
5
4
for a single variable function, which (c) h (d) h
is
[GATE-2009 (PI)]
(a) linear (b) parabolic −
105. The differential equation dx = 1 x
(c) logarithmic (d) hyperbolic dt 
is discretised using Euler’s numerical
[GATE-2016-ME-SET II]
integration method with a time step
102. P(0, 3), Q (0.5, 4) and R (1, 5) are  T  0. What is the maximum

three points on the curve defined by permissible value of T to ensure
f(x). Numerical integration is carried stability of the solution of the
out using both Trapezoidal rule and corresponding discrete time equation?

Simpson’s rule within limits x = 0

and x = 1 for the curve. The (a) 1 (b) /2
difference between the two results (c)  (d) 2
will be
[GATE-2011]
(a) 0 (b) 0.25
106. The ordinary differential equation
(c) 0.5 (d) 1 dx = − 3x + , with x (0) = 1 is to be
2
[GATE-2017 ME SESSION-I] dt
solved using the forward Euler
FORWARD EULER METHOD method. The largest time step that can

du be used to solve the equation without
103. Consider the equation = 3t + 2 1 making the numerical solution
dt unstable is ________.

with u = 0 at t = 0. This is
numerically solved by using the [GATE-2016-EC-SET 2]
forward Euler method with a step
size. t = 2. The absolute error in 107. Variation of water depth (y) in a
the solution in the end of the first gradually varied open channel flow is
time step is _____. given by the first order differential
equation
[GATE-2017]
− 10 ln y
−
104. During the numerical solution of a dy = 1 e 3
−
first order differential equation using dx 250 45e − 3ln y
the Euler (also known as Euler Given initial condition y (x = 0) = 0.8
Cauchy) method step size h, the local m. The depth (in m, up to three
truncation error is of the order of

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