GATE (Industrial Engineering Notes)

Part 1 : Metrology and Quality Control

Part 2 : Industrial Engineering

Part 3 : Queuing Theory, Simplex Method, Transportation and Assignment Problems

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Numerical Solutions to Ordinary Differential Equations

We will discuss the following methods:


Taylor’s Series Method

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Euler’s Method and Modified Euler’s Method

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Runge Kutta 2nd Order Method

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Runge Kutta 4th Order Method

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Numerical Integration

Techniques of numerical integration have been developed because of the following reasons:

  1. Integration, in general, is a difficult operation.
  2. We may have experimental data, but we may not know the actual function generating those values.

In this article, we will discuss 3 commonly used numerical integration techniques, the trapezoidal rule, Simpson’s 1/3rd rule and Simpson’s 3/8th rule.


Trapezoidal Rule

  • Uses the single degree curve for integration

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Simpson’s 1/3rd Rule

This rule is also known as Simpson’s rule.

  • Uses a second degree curve for integration.

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Simpson’s 3/8th Rule

  • It uses a third degree curve for integration

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Numerical Solutions of Non-Linear Algebraic Equations

Many real world applications involve non-linear equations, such as {x cos (x) - 1 = 2}. In most of the cases, such equations cannot be solved analytically. Numerical methods discussed below come to our rescue. These methods provide an approximate answer to the desired level of accuracy.


Bisection Method

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Newton-Raphson Method

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Regula Falsi/ False Position/ Modified Secant Method

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Secant Method

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