The differential equation is to be solved using the backward (implicit) Euler's method with the boundary condition y = 1 at x = 0 and with a step size of 1. What would be the value of y at x = 1?
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Engineering Maths · all questionsNumerical Methods
Numerical Methods
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What is the value of (1525)0.2 to 2 decimal places?
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Numerical integration using trapezoidal rule gives the best result for a single variable function, which is
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Solution of the variables x1 and x2 for the following equations is to be obtained by employing the Newton-Raphson iterative method
equation (i) 10x2 sin x1 - 0.8 = 0
equation (ii) 10 - 10x2 cos x1 - 0.6 = 0
Assuming the initial values x1 = 0.0 and x2 = 1.0, the Jacobian matrix is
equation (i) 10x2 sin x1 - 0.8 = 0
equation (ii) 10 - 10x2 cos x1 - 0.6 = 0
Assuming the initial values x1 = 0.0 and x2 = 1.0, the Jacobian matrix is
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The root of the function f(x) = x3 + x - 1 obtained after first iteration on application of Newton Raphson scheme using an initial guess of x0 = 1 is
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The velocity v (in m/s) of a moving mass, starting from rest, given as Using Euler forward difference method (also known as Cauchy-Euler method) with a step size of 0.1 s, the velocity at 0.2 s evaluate to
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Let x2 - 117 = 0. The iterative steps for the solution using Newton-Raphson's method is given by
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A piecewise linear function f(x) is plotted using thick solid lines in the figure below (the plot is drawn to scale).
If we use the Newton-Raphson method to find the roots of f(x) = 0 using x0, x1 and x2 respectively as initial guesses, the roots obtained would be
If we use the Newton-Raphson method to find the roots of f(x) = 0 using x0, x1 and x2 respectively as initial guesses, the roots obtained would be
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Function f is known at the following points
The value of computed using the continuous at x = 3?
| x | 0 | 0.3 | 0.6 | 0.9 | 1.2 | 1.5 | 1.8 | 2.1 | 2.4 | 2.7 | 3.0 |
| f(x) | 0 | 0.09 | 0.36 | 0.81 | 1.44 | 2.25 | 3.24 | 4.41 | 5.76 | 7.29 | 9.00 |
The value of computed using the continuous at x = 3?
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The integral when evaluated by using Simpson's 1/3 rule on two equal subintervals each of length 1, equals
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In numerical integration using Simpson's rule, the approximating function in interval is a
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Using Simpson's 1/3 rule for numerical integration, the consecutive points are joined by a
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The differential equation with x(0) = 0 and the constant > 0, is to be numerically integrated using the forward Euler method with a constant integration time step T. The maximum value of T such that the numerical solution of x converges is
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Match the following:
| List-I | List-II |
| P. 2nd order differential equation | 1. Runge-Kutta Method |
| Q. Non-linear algebraic equation | 2. Newton-Raphson Method |
| R. Linear algebraic equation | 3. Gauss Elimination |
| S. Numerical integration | 4. Simpson's rule |
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Match the application to appropriate numerical method.
| Application | Numerical Method |
| P1: Numerical integration | M1: Newton-Raphson Method |
| P2: Solution to a transcendental equation | M2: Runge-Kutta Method |
| P3: Solution to a system of linear equations | M3: Simpson's 1/3-rule |
| P4: Solution to a differential equation | M4: Gauss Elimination Method |
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The estimate of obtained using Simpson's rule with three-point function evaluation exceeds the exact value by
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The integral with x2 > x1 > 0 is evaluated analytically as well as numerically using a single application of the trapezoidal rule. If is the exact value of the integral obtained analytically and J is the approximate value obtained using the trapezoidal rule, which of the following statements is correct about their relationship?
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The error in for a continuous function estimated with h = 0.03 using the central difference formula is 2 × 10-3. The values of x0 and f(x0) are 19.78 and 500.01, respectively. The corresponding error in the central difference estimate for h = 0.02 is approximately
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