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Prof. Muhammad Saeed ( Differentiation and Integration )

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2M.Sc. Physics 1.Differenti ation First Derivative Central Difference

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M.Sc. Physics3 Second Derivative Central Difference Third Derivative Averaged Difference Fourth Derivative Central Difference

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M.Sc. Physics4 2. Integ ration Newton-Cotes Formulas i. ii. iii. Trapezoidal Rule Simpsons 1/3 Rule

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M.Sc. Physics5 Simpsons 3/8 Rule Gaussian Quadrature i)Two-Term Formula ii) Three-Term Formula iii) Four-Term Formula

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M.Sc. Physics6 Integral Change for Gaussian Quadrature iv) Five-Term Formula Legendre Polynomials The roots of Legendre Polynomials ( ) are t values in Gaussian Quadrature. n corresponds to nth term formula.

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M.Sc. Physics7 Rombergs Integration More accurate value is the result of smaller h(interval) and n is the order of error in terms of h. Steps: i)Estimate integral for a higher value of h ii)Then for h/2, h/4, h/8 ………. iii)Use n=2 and apply Romberg Integration iv)Apply Romberg Integration on the new values taking n=4 v) Similarly for next calculation take n=6, so on and so forth. n=2, 4, 6, 8, 10, …………… Adaptive Scheme of Integration Make h smaller where rate of change is greater. Multiple Integration Improper Integrals

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M.Sc. Physics8

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NUMERICAL DIFFERENTIATION AND INTEGRATION

NUMERICAL DIFFERENTIATION AND INTEGRATION

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