An Efficient Method of Bounded Solution of a System of Differential Equations Using Linear Legendre Multi Wavelets

8/10/2019 An Efficient Method of Bounded Solution of a System of Differential Equations Using Linear Legendre Multi Wavelets

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18 Meenu Devi, S. R. Verma & M. P. Singh

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(2.1)

Where are dilation and translation parameters respectively.

*Corresponding author:

E-mail address: [email protected] (S. R. Verma).

If we restrict the parameters to discrete values i.e. [3], we have

(2.2)

Let be a function of space. It said to be scaling function for if it satisfy the following condition

. (2.3)

The nested sequence of the subspaces of with scaling function is formed multi resolution

analysis (MRA) [3].

For any orthogonal MRA with a multi scaling function . There exists multi wavelet function orthogonal to

each other, given by [3]:

(2.4)

And form an orthonormal basis for certain condition. For construction the linear Legendre

multi wavelet, firstly the scaling functions are defined as following:

(2.5)

By the definition of MRA,

(2.6)

We construct the Linear Legendre multi-wavelet by translating and dilating the mother wavelet and ψ are given

by

(2.7)

The family form an orthonormal basis for and subfamily j

nk ,ψ where 10, jand,...2,1,0 ==k is

an orthonormal for ]1,0[2 L .



An Efficient Method of Bounded Solution of a System of 19 Differential Equations Using Linear Legendre Multi Wavelets

www.tjprc.org [email protected]

3. METHOD OF THE SOLUTION OF SYSTEM OF HOMOGENEOUS LINEAR DIFFERENTIAL

EQUATIONS

Consider the system [18]:

(3.1)

(3.2)

With initial conditions ,

Where are constants. After approximation with

the help of function approximation and Operational matrix of Integration [12, 14, 19, 20, 21], one can get

(3.3)

(3.4)

Thus, by using equations (3.3) and (3.4), the equations (3.1) and (3.2) reduce into the equations (3.5) and (3.6)

respectively

Or

i.e. (3.5)

.

Similarly , (3.6)

Where

One can get the values of from the equations (3.5) and (3.6) with the help of ref. [22] and putting these

values in equation (3.4) we obtained .

Theorem 3.1: Let is the exact solution of the system [18].







(3.1.7)

If for and , then one has

. (3.1.8)

Corollary 3.2 If , any approximate solution of the system , where [0,1]

will satisfy

.

Proof: Taking in equation (3.1.8), we get

.

4. ILLUSTRATIVE EXAMPLES

Example 4.1. Consider the system of homogeneous linear differential equations

0 (4.1.1)

0 (4.1.2)

With initial conditions

Approximating the unknown functions , and , we have

(4.1.3)

Where , , is operational matrix of integration and LLMW bases

by ref. [12, 14].

After using equation (4.1.3), the equations (4.1.1) and (4.1.2) resulted into the following forms respectively

(4.1.4)

(4.1.5)

From equation (4.1.5), we get

(4.1.6)

With this value of , equation (4.1.4), gives





(4.1.7)

The simplification of the equations (4.1.6) and (4.1.7) yields:

,

With these values of , one can get from equation (4.1.3)

,

The exact and approximate solutions are depicted in Figure 1 and Figure 2:

exact solutionapproximate solution

0.2 0.4 0.6 0.8 1.0

2.0

1.5

1.0

Figure 1:

exact soluti on

approximate solution

0.2 0.4 0.6 0.8 1.0

1.5

2.0

2.5

Figure 2:





And absolute error of exact and approximate solutions is given in Table 1:

Table 1

tExact

SolutionApproximate

SolutionError

Exact

SolutionApproximate

SolutionError

0.0 -1.0000 -0.994083 5.9 1.0000 0.994083 5.90.1 -1.10517 -1.10769 2.5 1.10517 1.10769 2.50.2 -1.2214 -1.2213 0.1 1.2214 1.2213 0.10.3 -1.34986 -1.34936 0.4 1.34986 1.34936 0.40.4 -1.49182 -1.49524 3.4 1.49182 1.49524 3.40.5 -1.64872 -1.63896 9.7 1.64872 1.63896 9.70.6 -1.82212 -1.82627 4.1 1.82212 1.82627 4.10.7 -2.01375 -2.01358

0.12.01375 2.01358

0.10.8 -2.22554 -2.22472 0.8 2.22554 2.22472 0.80.9 -2.4596 -2.46523 5.6 2.4596 2.46523 5.61.0 -2.71828 -2.56347 15 2.71828 2.56347 15

Example.4.2. Consider the system of homogeneous linear differential equations

(4.2.1)

(4.2.2)

With initial condition

Likewise the example 4.1, one can get

Y

And

,

.





The exact solutions and approximate solutions are traced in Figure 3 and Figure 4:

exact solut ion


0.2 0.4 0.6 0.8 1.0

.5

1.0

1.5

2.0

2.5

Figure 3 :

exact solution


0.2 0.4 0.6 0.8 1.0

1.5

2.0

2.5

Figure 4 :

Absolute error of exact and approximate solution:

Table 2

tExact

SolutionApproximate

Solution

ErrorExact

SolutionApproximate

Solution

Error

0.0 0.000000 0.0126046 12 1.0000 0.994083 5.9

0.1 0.110517 0.115794 5.2 1.10517 1.10769 2.50.2 0.244281 0.244193 0.08 1.2214 1.2213 0.10.3 0.404958 0.403581 1.3 1.34986 1.34936 0.40.4 0.59673 0.604916 8.1 1.49182 1.49524 3.40.5 0.824361 0.798677 2.5 1.64872 1.63896 9.70.6 1.09327 1.10402 10 1.82212 1.82627 4.10.7 1.40963 1.40937 0.2 2.01375 2.01358 0.10.8 1.78043 1.77772 2.7 2.22554 2.22472 0.80.9 2.21364 2.22992 16 2.4596 2.46523 5.61.0 2.71828 2.30975 40 2.71828 2.56347 15





5. CONCLUSIONS

In this paper, a well- organized method for solving system of homogeneous linear differential equations is

derived. Two examples are solved by this method and got more accurate solutions which are depicted by graphs because

the exact and approximate solutions are all most over lapping and solutions are bounded too. The applications of system of

homogeneous differential equations are cascade model, Newton cooling model etc.

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An Efficient Method of Bounded Solution of a System of Differential Equations Using Linear Legendre Multi Wavelets