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Chapter: Mathematics (maths) - Numerical Differentiation and Integration

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Numerical Differentiation and Integration

1 Numerical Differentiation Derivatives using divided differences Derivatives using finite Differences Newton`s forward interpolation formula Newton`s Backward interpolation formula 2 Numerical integration Trapezoidal Rule Simpson`s 1/3 Rule Simpson`s 3/8 Rule Romberg`s intergration 3 Gaussian quadrature Two Point Gaussian formula & Three Point Gaussian formula 4 Double integrals Trapezoidal Rule & Simpson`s Rule

NUMERICAL DIFFERENTIATION AND INTEGRATION

 

 

1 Numerical Differentiation Derivatives using divided differences

Derivatives using finite Differences

Newton`s forward interpolation formula

Newton`s Backward interpolation formula

 

2 Numerical integration

Trapezoidal Rule

Simpson`s 1/3 Rule

Simpson`s 3/8 Rule

Romberg`s intergration

 

3 Gaussian quadrature

Two Point Gaussian formula & Three Point Gaussian formula

 

4 Double integrals

Trapezoidal Rule & Simpson`s Rule

 

 

 

Introduction

 



For some particular value of x from the given data (xi,yi), i=1, 2, 3…..n where y=f(x) i explicitly. The interpolation to be used depends on the particular value of x which derivatives are required. If the values of x are not equally spaced, we represent the function by difference formula and the derivatives are obtained. If the values of x are equally spaced, the derivatives are calculated by using Newton’s Forwardinterpolationorformulabackwa.Ifthe derivatives are required at a point near the beginning of the table, we use Newton’sForward interpolation formula and if the derivatives are required at a point near the end of table. We use backward interpolation formula.

 

 

 

 

1  Derivatives using divided differences

 

Principle

 

First fit a polynomial for the given difference data interpolation using Newton ’s divided difference interpolation formula and compute the derivatives for a given x.

 

Problems

 

1.Compute  f(3.5) and f ‘’ (4) given that f(0)=2,f(1)=3;f(2)=12 and f(5)=147

Sol:

 

Divided difference table



 

2.Find the values f’ (5)and f’’ (5) using the following data

x    0    2  3  4  7 9

F(x)   4    26  58  112  466 22


Ans: 98 and 34.

 

1.1  Derivatives using Finite differences

Newton’s difference Forward Formula



Problems

 

1.Find the first two derivatives of y at x=54 from the following data

x        50               51               52                                  53               54

y        3.6840                  3.7083                  3.7325                           3.7563        3.7798


Sol:

Difference table & By     Newton’s   Backward   difference   formula



 

2.Find first and second derivatives of the function at the point x=12 from the following data


x                 1                 2                 3                 4                 5       

y                 0                 1                 5                 6                 8       

Sol:

Difference table


 



 

2 Numerical integration

 

(Trapezoidal rules, Romberg & Simpson’s integration) Introduction The process of computing the value of a definite integral from a set of values (xi,yi),i=0,1,2,x=a;…..xb of Where the function y=f(x) is called Numerical integration.

 

Here the function y is replaced by an interpolation formula involving finite differences and then integrated between the limits a and b, the value is found.

 

General Quadrature formula for equidistant ordinates (Newton formula)cote’s

 

On simplification we obtain

 


This is the general Quadrature formula

By putting n=1, Trapezoidal rule is obtained

By putting n=2,   Simpson’s 1/3 rule is derived

By putting n=3,   Simpson’s 3/8 rule is derived.

 

Note


 

Simpson’s   1/3   rule


 

Error in   Simpson’s 1/3   rule


 

Note

 

The Error of this formula is of order h5 and the dominant term in the error is given by 


 

 

Romberg’s integration

 

A simple modification of the Trapezoidal rule can be used to find a better approximation to the value of an integral. This is based on the fact that the truncation error of the Trapezoidal rule is nearly proportional to h2.


This value of I will be a better approximation than I1 or I2.This method is called Richardson’s deferred approach to the limit.

If h1=h and h2=h/2,then we get


 


 

 






3  Gaussian Quadrature

For evaluating the integral 


 , we derived some integration rules which require the values of the function at equally spaced points of the interval. Gauss derived formula which uses the same number of function values but with the different spacing gives better accuracy.


Where  wi  and u i  are called the weights and abscissa respectively. In this formula, the abscissa and weights are symmetrical with respect to the middle of the interval. The one-point Gaussian Quadrature formula is given by


 

, which is exact for polynomials of degree upto 1.

 

Two point Gaussian formula

 


And this is exact for polynomials of degree upto 3.

 

Three point Gaussian formula


Which is exact for polynomials of degree upto 5

Note:


 

Error terms

The error in two –point Gaussian formula=


 





 

4 Double integrals

In this session we shall discuss the evaluation of


 

Trapezoidal and Simpson’s rule.

The formulae for the evaluation of a double integral can be obtained by repeatedly applying the

Trapezoidal and   simpson’s   rules.

Consider the integral I  


The integration in (1) can be obtained by successive application of any numerical integration formula with respect to different variables.

 

Trapezoidal Rule for double integral


 

Simpson’s rule   for   double   integral


 

 

problems



 


 

Tutorial problems

Tutorial -1

1.Find the value of  f ‘’(3) using divided differences, given data

                                x     0                 1                 4        5       

                                F(x)          8                 11               68      123   

Ans:16

 

2.Find the first and second derivatives of the function at the point x=1.2 from the following data

 

X                             1     2        3        4        5

Y                             0     1        5        6        8

Ans:14.17                                        

 

3.A rod is rotating in a plane .The angle θ ( values of time t (seconds) are given below

 

 

t                               0     0.2     0.4     0.6     0.8     1.0     1.2

                                                                                     

θ                              0     0.122 0.493 1.123 2.022 3.220 4.666

                                                                                     

 

 

Find the angular velocity and angular acceleration of the rod when t=06 seconds.

 

Ans:6.7275 radians/sec2.

 

4.From the table given below ,for what value of x, y is minimum? Also find this value of y.

 

X                                    3                           4        5        6                 7        8                

Y                                    0.205          0.240 0.25   0.262          0.250 0.224         

Ans:5.6875,0.2628                                                                 

5.Find the first and second derivatives of y at x=500 from the following data   

                                                                                                                                                                               

x                                    500                       510    520    530             540    550            

y                                    6.2146                  6.2344        6.2538        6.2792                  6.2196                                6.3099              

Ans:0.002000,-0.0000040                                                                         

 

 


 

 


 

Part A

 

1. State the disadvantages of Taylor series method.

 

Sol:

 

In the differential equation = f(x,y) the function f(x,y) may have a complicated

 

algebraical structure. Then the evaluation of higher order derivatives may become tedious. This is the demerit of this method.

 

2.      Which   is   better   Taylor’s   method   or   R.K   met

 

Sol:

 

R.K methods do not require prior calculation of higher derivatives of y(x), as the Taylor method does. Since the differential equations using in application are often complicated, the

 

 

 

calculation of derivatives may be difficult. Also R.K formulas involve the computations of f(x,y) at various positions instead of derivatives and this function occurs in the given equation.

 

3. What is a predictor- collector method of solving a differential equation?

 

Sol:

predictor- collector methods are methods which require the values of y at xn, xn-1, xn-2,…

for computing the values of y at xn+1. We first use a formula to find the values of y at xn+1 and this is known as a predictor formula. The value of y so get is improved or corrected by another formula known as corrector formula.

 

4. Define a difference Quotient.

 

Sol:

 

A difference quotient is the quotient obtained by dividing the difference between two values of a function, by the difference between the two corresponding values of the independent.

 


 



3.Find the first and second derivative of the function tabulated below at x=0.6

x        0.4     0.5     0.6     0.7     0.8

y        1.5836        1.7974        2.0442        2.3275        2.6511

 

4. Using Romberg’s method dxcorrectto4 decimal to places compute. Also evaluate the same integral using three –point Gauss quadrature formula. Comment on the obtained values by comparing with exact value of the integral which is equal to .

7.Evaluate I=    by dividing the range into ten equal parts,using                                                 

(i)Trapezoidal rule                                                                                                                                                                  

(ii)Simpson’s –third rule.

Verify one your answer with actual integration.                                                 

8. Find the first two derivatives of at x=50 and x=56, for the given table :                                      

                                                                  

x        50                         51                         52               53               54               55               56                  

Y=x1/3         3.6840                  3.7084                  3.7325        3.7325        3.7798        3.8030          3.8259                 

9.The velocities of a car running on a straight road at intervals of 2 minutes are given below:                  

Time(min)            0                 2                 4                 6                 8                 10               12                  

Velocity(km/hr)    0                 22               30               27               18               7                 0                  

                             rd                                                                                                                                  

Using   Simpson’s-rulefindthedistance1/3covered by the car.                                                

 

10. Given   the   following   data,   find   y’(6)   and   the

x        0        2        3        4        7        9

Y       4        26      58      112    466    922


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