Solving Variational Problems Using Haar Wavelet J. Kumar

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ISSN 2319-8133 (Online)

(An International Research Journal), www.compmath-journal.org

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2

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1,2,3

(Received on: January 15, 2016) ABSTRACT

Haar wavelets are used for approximating solution of variational problems of calculus of variations. The variational problem is converted into differential equation using Euler Lagrange's equation and then Haar collocation method is applied to solve this differential equation.

Mathematical Subject Classification: 65L60; 42C40.

Keywords: Variational problem, Euler Lagrange's equation, Collocation method, Haar wavelet.

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0 0 0

times

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i i

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1, 1,

1 1

M M

i i i i

i i

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1, 2,

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i i

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1

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The numerical solution and analytic solution are compared in table 1.

Table 1: Estimated and analytic solution of

for Example 4.1

Example 4.2 Consider the problem of finding the minimum of the functional

t Present method Analytic solution 0.0625

0.1875 0.3125 0.4375 0.5625 0.6875 0.8125 0.9375

0.030273437500000 0.084960937500000 0.131835937500000 0.170898437500000 0.202148437500000 0.225585937500000 0.241210937500000 0.249023437500000

0.030273437500000 0.084960937500000 0.131835937500000 0.170898437500000 0.202148437500000 0.225585937500000 0.241210937500000 0.249023437500000

(7)

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1

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2 2

1, 2,

1 1

M M

i i i i

i i

2

1, 2,

1

M

i i i i

i

2 1

2

t t t t

The numerical solution and analytic solution are compared in table 2.

Table 2: Estimated and analytic solution of

for Example 4.2

1 2 0

I   y t  ty t dt

t Present method Analytic solution 0.0625

0.1875 0.3125 0.4375 0.5625 0.6875 0.8125 0.9375

0.026646295899747 0.074485209357164 0.115662443117627 0.150822650547252 0.180516284991315 0.205208217439155 0.225285014347883 0.241060989563427

0.026777937796972 0.074883846791318 0.116338662375549 0.151790959882946 0.181795403116672 0.206821422262240 0.227260558289723 0.243432588751479

(8)

1

y t

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1

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M

i i

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0

i i

2 0, 1

M

i i

i

2 1, 1

2 2

0, 1,

1 1

M

i i

i

M M

i i i i

i i

2 2

0, 2,

1 1

M M

i i i i

i i

2

1

M i i i

2

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The numerical solution and analytic solution are compared in table 3.

Table 3: Estimated and analytic solution of

for Example 4.3

1 2 0

1 1

1

y y t t

y t

2

1

M i i i

2 2 2 2 3 2

0, 1, 4, 0,

1 1 1 1

M M M M

i i i i i i i i

i i i i

Analytic solution for this problem is given by

t Present method Analytic solution 0.0625

0.1875 0.3125 0.4375 0.5625 0.6875 0.8125 0.9375

-0.000976562500000 -0.008789062500000 -0.024414062500000 -0.047851562500000 -0.079101562500000 -0.118164062500000 -0.165039062500000 -0.219726562500000

-0.000976562500000 -0.008789062500000 -0.024414062500000 -0.047851562500000 -0.079101562500000 -0.118164062500000 -0.165039062500000 -0.219726562500000

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4

3 2

The numerical solution and analytic solution are compared in table 4.

Table 4 : Estimated and analytic solution of

for Example 4.4

1 2 0

2

0

1

y t y t

2

1

M i i i

2 0, 1 1

M

i i

i



t Present method Analytic solution 0.0625

0.1875 0.3125 0.4375 0.5625 0.6875 0.8125 0.9375

-0.003746032714844 -0.030967712402344 -0.078900655110677 -0.141685485839844 -0.214439392089844 -0.293256123860677 -0.375205993652344 -0.458335876464844

-0.003746032714844 -0.030967712402344 -0.078900655110677 -0.141685485839844 -0.214439392089844 -0.293256123860677 -0.375205993652344 -0.458335876464844

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2 2, 2

M

i i

i

2

1

M i i i

i

2

2

2

The numerical solution and analytic solution are compared in table 5.

Table 5 : Estimated and analytic solution of

for Example 4.5

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11

9

11

The Authors are thankful to the University Grants Commission, India for providing the financial assistance for the preparation of the manuscript.

t Present method Analytic solution 0.0625

0.1875 0.3125 0.4375 0.5625 0.6875 0.8125 0.9375

0.001953125000000 0.017578125000000 0.041015625000000 0.025390625000000 -0.029296875000000 -0.076171875000000 -0.107421875000000 -0.123046875000000

0.001953125000000 0.017578125000000 0.041015625000000 0.025390625000000 -0.029296875000000 -0.076171875000000 -0.107421875000000 -0.123046875000000

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Quintic Spline Method, Journal of Hyperstructures, 3 (1), pp. 40-52 (2014).

References

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