8_Numerical Integration

8_Numerical Integration - Numerical Integration...

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Numerical Integration Dr.B.Santhosh Department of Mechanical Engineering Dr.B.Santhosh Department of Mechanical Engineering Numerical Integration
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Introduction Compute integraltext b a f ( x ) dx where f ( x ) is a given function Approximate the definite integral integraltext b a f ( x ) dx by the sum I = n summationdisplay i = 1 A i f ( x i ) x i are nodal abscissas and A i are the weights which depends on the particular rule used for integration Dr.B.Santhosh Department of Mechanical Engineering Numerical Integration
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Introduction If the points where the function f ( x ) evaluated (nodes) are evenly spaced in the interval, the term numerical integartion is used If the location of the nodes may be determined in order to optimize the method of integration in some way, the methods are known as quadrature The first class of methods are known as the Newton-Cotes formula Trapezoidal rule Simpson’s rule The second class of methods are known as Guassian Quadrature Dr.B.Santhosh Department of Mechanical Engineering Numerical Integration
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Applications in Engineering Integral Transforms - Laplace, Fourier, Hankel Special Functions - gamma, beta, Bessel, error functions, Fresnel and elliptic integrals Finite element and boundary element methods for pdes Integral equations and variational methods Probability and Statistics : Probability distributions, expectations and moments Classical and Quantum physics : potential and free energy of many systems Dr.B.Santhosh Department of Mechanical Engineering Numerical Integration
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Trapezoidal Rule Approximate the curve by a straight line between a and b integraldisplay b a f ( x ) dx h 2 [ f ( a )+ f ( b )] where h = b - a Dr.B.Santhosh Department of Mechanical Engineering Numerical Integration
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Composite Trapezoidal Rule To improve the trapezoidal method, apply the above method repeatedly on several sub-intervals. Consider two intervals [ a x 1 ] and [ x 1 b ] Here x 1 = a + b 2 and h = b a 2 integraldisplay b a f ( x ) dx = integraldisplay x 1 a f 1 ( x )+ integraldisplay b x 1 f 2 ( x ) h 2 [ f ( a )+ f ( x 1 )]+ h 2 [ f ( x 1 )+ f ( b )] integraldisplay b a f ( x ) dx b - a 4 [ f ( a )+ 2 f ( x 1 )+ f ( b )] Dr.B.Santhosh Department of Mechanical Engineering Numerical Integration
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Composite Trapezoidal Rule For n sub intervals h = b a n integraldisplay b a f ( x ) dx = integraldisplay x 1 a f ( x ) dx + .......... + integraldisplay b x n - 1 f ( x ) dx h 2 [ f ( a )+ f ( x 1 )]+ ........ + h 2 [ f ( x n 1 )+ f ( b )] integraldisplay b a f ( x ) dx = b - a 2 n [ f ( a )+ 2 f ( x 1 )+ ........ + 2 f ( x n 1 )+ f ( b )] Dr.B.Santhosh Department of Mechanical Engineering Numerical Integration
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Error in Trapezoidal Method The error involved in polynomial interpolation is given by f ( x ) - P n 1 ( x )= ( x - x 1 )( x - x 2 ) ...... ( x - x n ) n ! f ( n ) ( ξ ) where ξ is somewhere in the interval [ a b ] whose value is known integraldisplay b a f ( x ) dx I = h 2 [ f ( a )+ f ( b )] error E = integraldisplay b a f ( x ) dx - I Dr.B.Santhosh Department of Mechanical Engineering Numerical Integration
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Error in Trapezoidal Method E = 1 2 !
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