Numerical ODE Solutions - Numerical Methods - Engineering Sciences Previous Year Questions

1

2007 · Engineering Sciences · Numerical Methods · Numerical ODE Solutions

Engineering Sciences (XE) 2007

Choose the correct predictor scheme to solve the above initial value problem at x = 0.8 from the following

A

\(y_{n+1} = y_n + \frac{4h}{3}(2f_{n-1} - f_{n-2} + 2f_{n-3})\)

B

\(y_{n+1} = y_{n-3} + \frac{4h}{3}(2f_{n-2} - f_{n-1} + 2f_n)\)

C

\(y_{n+1} = y_{n-1} + \frac{h}{3}(4f_{n-1} - 5f_n + 4f_{n+1})\)

D

\(y_{n+1} = y_{n-3} + \frac{4h}{3}(2f_{n-1} - f_{n-2} + 2f_{n-3})\)

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2

2007 · Engineering Sciences · Numerical Methods · Numerical ODE Solutions

Engineering Sciences (XE) 2007

On solving the initial value problem
\(\frac{dy}{dx} = xy^2\), \(y(1) = 1\) by Euler's method, the value of \(y\) at \(x = 1.2\) with \(h = 0.1\) is

A

1.1000

B

1.1232

C

1.2210

D

1.2331

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3

2007 · Engineering Sciences · Numerical Methods · Numerical ODE Solutions

Engineering Sciences (XE) 2007

The local error of the following scheme
\(y_{n+1} = y_n - \frac{h}{12}(5y'_{n+1} + 8y'_n - y'_{n-1})\)
by comparing with the Taylor series \(y_{n+1} = y_n + h y'_n + \frac{h^2}{2!} y''_n + ...\) is

A

O(h^4)

B

O(h^3)

C

O(h^2)

D

O(h^5)

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4

2007 · Engineering Sciences · Numerical Methods · Numerical ODE Solutions

Engineering Sciences (XE) 2007

Using the correct predictor scheme from Q.27, the value of y(0.8) is

A

1.8680

B

1.8750

C

1.8890

D

1.9055

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5

2008 · Engineering Sciences · Numerical Methods · Numerical ODE Solutions

Engineering Sciences (XE) 2008

While solving the initial value problem \( \frac{dy}{dx} + ky = 0, \quad y(0) = 1 \) at \( x = h \) by fourth order Runge-Kutta method, the expression for \( k_3 \) is

A

\(-kh + \frac{(kh)^2}{2!} - \frac{(kh)^3}{3!} \)

B

\(-kh + \frac{(kh)^2}{2} - \frac{(kh)^3}{3} \)

C

\(-kh + \frac{(kh)^2}{2} - \frac{(kh)^3}{4} \)

D

\(-k \left( 1 + \frac{h}{2} - \frac{h^2}{3} \right) \)

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6

2010 · Engineering Sciences · Numerical Methods · Numerical ODE Solutions

Engineering Sciences (XE) 2010

Given that \(\frac{dy}{dx}=1+y^2\), \(y(0)=0\), which one of the following is nearest to \(y(0.4)\) computed by Euler's method with step size of 0.2 ?

A

0.408

B

0.404

C

0.208

D

0.204

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7

2013 · Engineering Sciences · Numerical Methods · Numerical ODE Solutions

Engineering Sciences (XE) 2013

Using Euler's method to solve the differential equation \[ \frac{d y}{d x} = 2 \cos \left( \frac{4 \pi x}{3} \right) - y, \; y(0) = 1 \] with step-size \( h = 0.25 \), the value of \( y(0.5) \) is

A

1.3125

B

1.1875

C

1.125

D

1.0625

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8

2017 · Engineering Sciences · Numerical Methods · Numerical ODE Solutions

Engineering Sciences (XE) 2017

A student wants to numerically solve the linear 1-D advection equation \(\frac{\partial \varphi}{\partial t} + c \frac{\partial \varphi}{\partial x} = 0\), where \(c = 300\) m s^-1. The value of the maximum time-step the student can consider according to CFL criterion for a spatial resolution of 3 km is

A

15 s

B

10 s

C

25 s

D

20 s

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9

2026 · Engineering Sciences · Numerical Methods · Numerical ODE Solutions

Engineering Sciences (XE) 2026

The initial value problem
\[ \frac{du}{dt} = u^2 + t^2, \quad t \geq 0, \] with \( u(0) = 1 \),
is solved by using the explicit Euler method with step size \( h = 0.2 \). Then, the value of \( u(0.4) \) (rounded off upto two decimal places) is __________

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