GATE Questions & Answers of Numerical Methods Mechanical Engineering

An explicit forward Euler method is used to numerically integrate the differential equatio

                                                                  $ \frac{dy}{dt}=y $

using a time step of 0.1. With the initial condition $ y\left(0\right)=1 $ , the value of $ y\left(1\right) $ computed by this method is ___________ (correct to two decimal places).

P (0,3), Q (0.5, 4), and R (1, 5) are three points on the curve defined by $\style{font-family:'Times New Roman'}{f(x)}$. Numerical integration is carried out  using both Trapezoidal rule and Simpson's rule within limits x=0 and x=1 for the curve. The difference between the two results will be

Usiing the trapezoidal rule, and dividing  the interval of integration into three equal subintervals, the definite integral $\int\limits_{-1}^{+1}\left|x\right|\;\mathrm{dx}$ is_______

The value of $\int\limits_{2.5}^4\ln\left(x\right)dx$ calculated using the Trapezoidal rule with five subintervals is _______

The definite integral 131xdx is evaluated using Trapezoidal rule with a step size of 1. The correct answer is _______

The real root of the equation 5x − 2cosx −1 = 0 (up to two decimal accuracy) is _______

Consider an ordinary differential equation dxdt=4t+4 If = x0 at t = 0, the increment in x calculated using Runge-Kutta fourth order multi-step method with a step size of Δt = 0.2 is

Match the CORRECT pairs.

Numerical Integration Scheme Order of Fitting Polynomial
P. Simpson’s 3/8 Rule 1. First
Q. Trapezoidal Rule 2. Second
R. Simpson’s 1/3 Rule 3. Third

The integral $\int\limits_1^3\frac1xdx$ when evaluated by using Simpson’s 1/3 rule on two equal subintervals each of length 1, equals

Torque exerted on a flywheel over a cycle is listed in the table. Flywheel energy (in J per unit cycle) using Simpson’s rule is

Angle (degree) 0 60 120 180 240 300 360
Torque (Nm) 0 1066 -323 0 323 -355 0

A calculator has accuracy up to 8 digits after decimal place. The value of $\int\limits_0^{2\pi}\sin x\operatorname{d}x$ when evaluated using this calculator by trapezoidal method with 8 equal intervals, to 5 significant digits is