Meixner Wavelet Method

Meixner Wavelet Method,we proposed a new algorithm by inserting Meixner polynomials in traditional Legendre wavelets method. This technique is successively applied to find the exact solution singular ordinary differential equations. The proposed technique is very simple and highly compatible for solving such kind of problems.

5.2.1 Methodology

5.2.1.1 Meixner Polynomials

Recurrence formula of Meixner polynomials is defined as

$\sum _{k=0}^n\frac{-1^k.n!)}{(k!.(n-k)!)}.\ \frac{x!}{(k!.(x-k)!}.\ \frac{(k!.Γ(x-β+1)}{(Γ(x-β-n+k+1))}.g^{-k}$ (5.1)

few Meixner polynomials given below

$\begin{array}{l}M_0=1,\\ M_1=\frac{1}{2x-1},\ \ \\ M_2=\frac{1}{4}\ x^2-\frac{9}{4}x+2,\\ M_3=\frac{1}{8}\ x^3-\frac{21}{8}\ x^2+\frac{17}{2}\ x-6,\end{array}$

5.2.1.2 Wavelets and Meixner Wavelets

In recent years, wavelets have found their way into many different fields of science and engineering [1, 2 and 3]. Wavelets constitute a family of functions constructed from the dilation and translation of a single function called the mother wavelet. When the dilation parameter a and the translation parameter b vary continuously, we have the following family of continuous wavelets:

$\psi_{a,b}(t)=|a|^{-1/2}\psi (\frac{t-b}{a})\ \ a,b∈R,\ \ a\ne 0.$ (5.2)

Meixner wavelets $\psi _nm(t)=\psi (k,n,m,t)$ have four arguments; n,k can assume any positive integer,m is the order for Meixner polynomials and t is the normalized time. They are defined on the interval [0, 1) by;

$\begin{array}{l}\ \ \ \ \ \psi _{n,m}(t)=\ \ \ \ \sqrt{m+\frac{1}{2}}s^{\frac{k}{2}}M_m\left(2^kt\ -\left(2n-1\right)\right),\ \ \ \ \ \ \ \ \frac{n}{2^k}\le t\le \ \frac{n}{2^{k+1}}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ otherwise\end{array}$ (5.3)

where m=0,1,2,…,M and $n=0,1,2,...,2^{k-1}$ The coefficient $\ m+\frac{1}{2}$ is for orthonormality. Here $M_m\left(t\right)$ are the well-known Meixner polynomials of order m which have been previously described.

5.2.1.3 Functions Approximation

A function f(t) define over [0,1) may be expanded as

$f(t)=\sum _{n=1}^{\infty }\sum _{m=0}^{\infty }c_{nm}\psi _{nm}(t),$ (5.4)

where $c_{nm}=〈f(t),\psi _{nm}(t)〉,$ in which $〈∙,∙〉$ denotes the inner product. If the infinite series in Eq. (5.4) is truncated, then Eq. (5.4) can be written as

$f(t)=\sum _{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}c_{nm}\psi _{nm}(t).$ (5.5)

It can be written as

$f(t)=\sum _{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}c_{nm}\psi _{nm}(t)=C^T\psi (t)$ (5.6)

where C and $\psi (t)$ are $2^{k-1}M\times 1$ matrices given as

$C=\left[\begin{matrix}c_{10},c_{11},c_{12},c_{13},...c_{1M-1},c_{20},c_{21},...\\ c_{2M-1},...,c_{2^{k-1}0},...,,c_{2^{k-1}M-1},...\end{matrix}\right]^T$ (5.7)

and,

$\psi =\left[\begin{matrix}\psi _{10},\psi _{11},\psi _{12}\psi _{13},...,\psi _{1M-1},\psi _{20},\psi _{21},...\\ \psi _{2M-1},...,\psi _{2^{k-1}0},...,,\psi _{2^{k-1}M-1},...\end{matrix}\right]^T$ (5.8)

5.3 Applications

5.3.1 Volterra Integral Equation

Consider the following Volterra integral equation [4] as follow

$y(x)=\sin ⁡(x)+\cos ⁡(x)+2\int _0^x\sin ⁡(x-t)u(t)dt\ ,$ (5.9)

The exact solution of Eq. (5.9) is

$y(x)=\exp ⁡(x).$ (5.10)

According to the Meixner Wavelets Method , we assume the trial solution

$y(x)=\sum _{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}c_{nm}ψ_{nm}(x)=C^T\Psi (x).\$ (5.11)

where C and $\psi (t)$ are $2^{k-1}M\times 1$ matrices given as

$C=\left[\begin{matrix}c_{10},c_{11},c_{12},c_{13},...,c_{1M-1},c_{20},c_{21},....c_{2M-1},...,c_{2^{k-1}M-1}\end{matrix}\right]^T$

$\Psi =\left[\begin{matrix}\psi _{10},\psi _{11},\psi _{12},\psi _{13},...,\psi _{1M-1},\psi _{20},\psi _{21},...,\psi _{2M-1},...,\psi _{2^{k-1}M-1}\end{matrix}\right]^T$

We apply the proposed technique to solve Eq. (5.9) with K=1 and M=10,we have Eq. (5.11) is

$y(x)=\sum _{n=1}^1\sum _{m=0}^2c_{nm}\psi _{nm}(x),=c_{10}\psi _{10}(x)+c_{11}\psi _{11}(x)+c_{12}\psi _{12}(x)+⋯$

$\ y\left(x\right)=C^T\Psi (x),$ (5.12)

where

$\psi _{nm}\left(x\right)=\sqrt{(}m+\frac{1}{2})\ 2^{\frac{k}{2}}M_m(2^kt-(2n-1)).$ (5.13)

From Eq. (5.13) we obtained

$y(x)=c_{10}+4\sqrt{3}xc_{11}-3\sqrt{3}c_{11}+16\sqrt{5}c_{12}x^2-28\sqrt{5}c_{12}x+12\sqrt{5}c_{12}+\dots$

$\ y\left(x\right)=C^T\Psi (x),$

where

$C=\left[c_{1,0},c_{1,1},c_{1,2},c_{1,3},c_{1,4,...}\right]^T$

and

$\Psi =\left[1,\sqrt{3}(4x-3),\sqrt{5}(12-28x+16x^2)+⋯\right]^T$

Substituting into the given problem we get

$C^T\Psi (x)=\sin ⁡(x)+\cos ⁡(x)+2\int _0^x\sin ⁡(x-t)C^T\Psi (t)dt,$ (5.14)

Substitute the collocating points are in Eq. (5.14), we have the system of equations

$x_i=\cos ⁡(\frac{(2i+1)\pi }{2^kM}),\ \ \ i=1,2,\dots ,9.$ (5.15)

Implementing the collocating points and imposing the initial value of the system, the matrix form is given as

$A_{10\times 10}C_{10\times 1}=b_{10\times 1}.$

After solving we get the following exact solution

$y(x)=1.0000++1.00000x+0.499999x^2+0.166667x^3+⋯.$

(a)

(b)

Fig. 5.1 (a)-(b) Comparison of Exact and Approximate Solution of Eq. (5.9) for M=10.

Table 5.1 Comparison of the Exact Solution and Approximate Solution of Eq. (5.9) for obtained from Meixner Wavelet Method

$\begin{tabular}{|c|c|c|c|} \hline X & Exact Solution & Approximate Solution & Error \\ \hline 0.0 & 1.000000000000000 & 1.000000000558292 & 5.58292E-10\\ \hline 0.1 & 1.105170918075648 & 1.105170918384351 & 5.08703E-10 \\ \hline 0.2 & 1.221402758160170 & 1.221402757932638 & 2.27532E-10\\ \hline 0.3 & 1.349858807576003 & 1.349858807027897 & 5.48106E-10\\ \hline 0.4 & 1.491824697641270 & 1.491824697334964 & 5.06306E-10 \\ \hline 0.5 & 1.648721270700128 & 1.648721271000500 & 5.00372E-10 \\ \hline 0.6 & 1.822118800390509 & 1.822118800975819 & 5.85310E-10 \\ \hline 0.7 & 2.013752707470477 & 2.013752707540811 & 7.03344E-11 \\ \hline 0.8 & 2.225540928492468 & 2.225540927922016 & 5.70452E-10 \\ \hline 0.9 & 2.459603111156950 & 2.459603111293860 & 1.36911E-10 \\ \hline \end{tabular}$

Now to solve Eq. (5.9) with K=1 and M=25, we have Eq. (5.11) is

$y(x)=\sum _{n=1}^1\sum _{m=0}^{24}c_{nm}\psi _{nm}(x)=C^T\Psi (x),$

where

$C=\left[c_{1,0},c_{1,1},c_{1,2},c_{1,3},...,c_{1,24}\right]^T$

and

$\Psi (t)=\left[\psi _{1,0},\psi _{1,1},\psi _{1,2},\psi _{1,3},...,\psi _{1,24}\right]^T$

Proceeding as before we have the matrix form

$A_{25\times 25}C_{25\times 1}=b_{25\times 1}.$

Proceeding as before we have the following graphical representation and error table.

(a)

(b)

Fig. 5.2 (a)-(b) Comparison of Exact and Approximate Solution of Eq. (5.9) for M=25.

Table 5.2 Comparison of the Exact Solution and Approximate Solution of Eq. (5.9) for M=25 obtained from Meixner Wavelet Method

$\begin{tabular}{|c|c|c|c|} \hline X & Exact Solution & Approximate Solution & Error \\ \hline 0.0 & 1.000000000000000 & 1.000000000000000 & 2.40000E-198\\ \hline 0.1 & 1.105170918075648 & 1.105170918075648 & 2.34685E-33 \\ \hline 0.2 & 1.221402758160170 & 1.221402758160170 & 5.64338E-33\\ \hline 0.3 & 1.349858807576003 & 1.349858807576003 & 5.90717E-33\\ \hline 0.4 & 1.491824697641270 & 1.491824697641270 & 2.86421E-33 \\ \hline 0.5 & 1.648721270700128 & 1.648721270700128 & 2.13898E-33 \\ \hline 0.6 & 1.822118800390509 & 1.822118800390509 & 1.27286E-33 \\ \hline 0.7 & 2.013752707470477 & 2.013752707470477 & 2.26925E-33 \\ \hline 0.8 & 2.225540928492468 & 2.225540928492468 & 5.43931E-33 \\ \hline 0.9 & 2.459603111156950 & 2.459603111156950 & 1.42910E-33 \\ \hline \end{tabular}$

Finally to solve Eq. (5.9) with K=1 and M=50, we have Eq. (5.11) is

$y(x)=\sum _{n=1}^1\sum _{m=0}^{49}c_{nm}\psi _{nm}(x)=C^T\Psi (x),$

where

$C=\left[c_{1,0},c_{1,1},c_{1,2},c_{1,3},...,c_{1,49}\right]^T$

and

$\Psi (t)=\left[\psi _{1,0},\psi _{1,1},\psi _{1,2},\psi _{1,3},...,\psi _{1,49}\right]^T$

Proceeding as before we have the matrix form

$A_{50\times 50}C_{50\times 1}=b_{50\times 1}.$

Proceeding as before we have the following graphical representation and error table.

(a)

(b)

Fig. 5.3 (a)-(b) Comparison of Exact and Approximate Solution of Eq. (5.9) for M=50.

Table 5.3 Comparison of the Exact Solution and Approximate Solution of Eq. (5.9) for obtained from Meixner Wavelet Method.

$\begin{tabular}{|c|c|c|c|} \hline X & Exact Solution & Approximate Solution & Error \\ \hline 0.0 & 1.000000000000000 & 1.000000000000000 & 7.09045E-42\\ \hline 0.1 & 1.105170918075648 & 1.105170918075648 & 7.01563E-42 \\ \hline 0.2 & 1.221402758160170 & 1.221402758160170 & 6.92844E-42\\ \hline 0.3 & 1.349858807576003 & 1.349858807576003 & 6.84144E-42\\ \hline 0.4 & 1.491824697641270 & 1.491824697641270 & 7.01018E-42 \\ \hline 0.5 & 1.648721270700128 & 1.648721270700128 & 7.17333E-42 \\ \hline 0.6 & 1.822118800390509 & 1.822118800390509 & 6.49724E-42 \\ \hline 0.7 & 2.013752707470477 & 2.013752707470477 & 2.11909E-42 \\ \hline 0.8 & 2.225540928492468 & 2.225540928492468 & 6.50203E-42 \\ \hline 0.9 & 2.459603111156950 & 2.459603111156950 & 5.13122E-42 \\ \hline 1.0 & 2.718281828459045 & 2.718281828459045 & 7.28792E-42 \\ \hline \end{tabular}$

5.3.2 Fredholm Integral Equations

Example 5.1 Consider the following integral equation [4] as follow

$y(x)=3⁡x+\exp ⁡(4x)+\frac{1}{16}(17+3\exp ⁡(4))+\int _0^1tu(t)dt,$ (5.16)

The exact solution of Eq. (5.16) is

$\ y(x)=3x+\exp ⁡(4x).$ (5.17)

According to the Meixner Wavelets Method , we assume the trial solution

$y(x)=\sum _{n=1}^{2^{k-1}}\sum _{m=0}^{M-1}c_{nm}\psi _{nm}(x)=C^T\Psi (x).$ (5.18)

where C and $\psi (t)$ are $2^{k-1}M\times 1$ matrices given as

$C=\left[\begin{matrix}c_{10},c_{11},c_{12},c_{13},...,c_{1M-1},c_{20},c_{21},....c_{2M-1},...,c_{2^{k-1}M-1}\end{matrix}\right]^T$

$\Psi =\left[\begin{matrix}\psi _{10},\psi _{11},\psi _{12},\psi _{13},...,\psi _{1M-1},\psi _{20},\psi _{21},...,\psi _{2M-1},...,\psi _{2^{k-1}M-1}\end{matrix}\right]^T$

We apply the proposed technique to solve Eq. (5.9) with K=1 and M=10,we have Eq. (5.11) is

$y(x)=\sum _{n=1}^1\sum _{m=0}^2c_{nm}\psi _{nm}(x),=c_{10}\psi _{10}(x)+c_{11}\psi _{11}(x)+c_{12}\psi _{12}(x)+⋯$

$\ y\left(x\right)=C^T\Psi (x),$ (5.19)

where

$\psi _{nm}\left(x\right)=\sqrt{(}m+\frac{1}{2})\ 2^{\frac{k}{2}}M_m(2^kt-(2n-1)).$ (5.20)

From Eq. (5.20) we obtained

$y(x)=c_{10}+4\sqrt{3}xc_{11}-3\sqrt{3}c_{11}+16\sqrt{5}c_{12}x^2-28\sqrt{5}c_{12}x+12\sqrt{5}c_{12}+\dots$

$\ y\left(x\right)=C^T\Psi (x),$

where

$C=\left[c_{1,0},c_{1,1},c_{1,2},c_{1,3},c_{1,4,...}\right]^T$

and

$\Psi =\left[1,\sqrt{3}(4x-3),\sqrt{5}(12-28x+16x^2)+⋯\right]^T$

Substituting into the given problem we get

$C^T\Psi (x)=3⁡x+\exp ⁡(4x)+\frac{1}{16}(17+3\exp ⁡(4))+\int _0^1tC^T\Psi (t)dt,$ (5.21)

Substitute the collocating points are in Eq. (5.21), we have the system of equations

$x_i=\cos ⁡(\frac{(2i+1)\pi }{2^kM}),\ i=1,2,\dots ,$ (5.22)

Implementing the collocating points and imposing the initial value of the system, the matrix form is given as

$A_{10\times 10}C_{10\times 1}=b_{10\times 10}.$

After solving we get the following exact solution

$y(x)=1.00077+7.00027x+7.96197x^2+⋯.$ (5.23)

Table 5.4 Comparison of the Exact Solution and Approximate Solution of Eq. (5.16) for obtained from Meixner Wavelet Method

$\begin{tabular}{|c|c|c|c|} \hline X & Exact Solution & Approximate Solution & Error \\ \hline 0.0 & 0.523598775598299 & 0.524440870663751 & 8.42095E-04\\ \hline 0.1 & 0.682364237868743 & 0.682866719602896 & 5.02482E-04 \\ \hline 0.2 & 0.843501108793284 & 0.843060729886135 &5.40379E-04\\ \hline 0.3 & 1.007584436725356 & 1.006423659770028 & 1.16078E-03\\ \hline 0.4 & 1.175397496610753 & 1.174653505097773 & 7.43992E-04 \\ \hline 0.5 & 1.348062078981481 & 1.348893691742286 & 8.31613E-04 \\ \hline 0.6 & 1.527295218001612 & 1.529275839392213 & 1.98062E-03 \\ \hline 0.7 &1.715985293814825 & 1.716296129005252 & 5.10835E-04 \\ \hline 0.8 & 1.919769514998634 & 1.916324286218898 & 5.44523E-03 \\ \hline 0.9 & 2.153235897503375 & 2.154570068363502 & 1.33417E-03 \\ \hline 1.0 & 2.570796326794897 & 2.500023913466260 & 7.07724E-02 \\ \hline \end{tabular}$

(a)

(b)

Fig. 5.4 (a)-(b) Comparison of Exact and Approximate Solution of Eq. (5.16) for M=10.

Now to solve Eq. (5.16) with K=1 and M=25, we have Eq. (5.18) is

$y(x)=\sum _{n=1}^1\sum _{m=0}^{24}c_{nm}\psi _{nm}(x)=C^T\Psi (x),$

where

$C=\left[c_{1,0},c_{1,1},c_{1,2},c_{1,3},...,c_{1,24}\right]^T$

and

$\Psi (t)=\left[\psi _{1,0},\psi _{1,1},\psi _{1,2},\psi _{1,3},...,\psi _{1,24}\right]^T$

Proceeding as before we have the matrix form

$A_{25\times 25}C_{25\times 1}=b_{25\times 1}.$

Proceeding as before we have the following graphical representation and error table.

Table 5.5 Comparison of the Exact Solution and Approximate Solution of Eq. (5.16) for obtained from Meixner Wavelet Method

$\begin{tabular}{|c|c|c|c|} \hline X & Exact Solution & Approximate Solution & Error \\ \hline 0.0 & 1.000000000000000 & 1.000000000000000 & 1.91402E-20\\ \hline 0.1 & 1.791824697641270 & 1.791824697641270& 2.99471E-18 \\ \hline 0.2 & 2.825540928492468 & 2.825540928492468 &5.89924E-18\\ \hline 0.3 & 5.220116922736547 & 5.220116922736547 &5.06079E-18\\ \hline 0.4 & 6.153032424395115 & 6.153032424395115 & 5.05414E-18 \\ \hline 0.5 & 8.889056098930650 & 8.889056098930650 & 2.67937E-18 \\ \hline 0.6 & 12.823176380641602 & 12.823176380641602 & 2.05210E-18\\ \hline 0.7 & 18.544646771097050 & 18.544646771097050 & 2.82739E-18 \\ \hline 0.8 & 26.932530197109349 & 26.932530197109349 & 5.29055E-18 \\ \hline \end{tabular}$

(a)

(b)

Fig. 5.5 (a)-(b) Comparison of Exact and Approximate Solution of Eq. (5.16) for M=25.

References

[1]. F.Mohammadi and M. M. Hosseini, Legendre wavelet method for solving linear stiff systems. Journal of Advanced Research in Differential Equations. 2 (1) 47-57. (2010)

[2]. F.Mohammadi, M. M. Hosseini. A comparative study of numerical methods for solving quadratic Riccati differential equations. Journal of Franklin Institute. 348, 156-164, 2011.

[3]. F.Mohammadi and M. M. Hosseini, A new Legendre wavelet operational matrix of derivative and its applications in solving the singular ordinary differential equations. Journal of Franklin Institute. 348, 1787-1796, 2011.

[4]. J.Y. Xiao, L. H. Wen, D. Zhang, Solving second kind integral equations by periodic wavelet Galerkin method. Applied Mathematics and Computation. 175, 508-518, 2006.