Araştırma Makalesi / Research Article Iğdır Üni. Fen Bilimleri Enst. Der. / Iğdır Univ. J. Inst. Sci. & Tech. 7(4): , 2017

Transcription

1 Araştırma Makalesi / Research Article Iğdır Üni. Fen Bilimleri Enst. Der. / Iğdır Univ. J. Inst. Sci. & Tech. 7(4): , 2017 Iğdır Üniversitesi Fen Bilimleri Enstitüsü Dergisi Iğdır University Journal of the Institute of Science and Technology Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials Faruk DÜŞÜNCELİ 1, Ercan ÇELİK 2 ABSTRACT: In this paper, the numerical solutions of complex differential equations are provided by the Hermite Polynomials and carried on two problems. As a result, the exact solutions and numerical one s have compared by tables and graphs that the method is practical, reliable and functional. Keywords: Hermite polynomials, linear complex differential equations, numerical solution Yüksek Mertebeden Lineer Kompleks Diferansiyel Denklemlerin Hermite Polinomları ile Nümerik Çözümleri Cilt/Volume: 7, Sayı/Issue: 4, Sayfa/pp: , 2017 ISSN: , e-issn: DOI: /jist ÖZET: Bu makalede lineer kompleks diferansiyel denklemleri hermite polinomları vasıtasıyla nümerik çözümünü sağladık ve iki test problemine uyguladık. Tam çözümler ile nümerik çözümleri tablo ve grafikler ile karşılaştırdık. Sonuç olarak metodumuzun güvenilir, pratik ve kullanışlı olduğunu gördük. Anahtar Kelimeler:Hermite polinomları, lineer kompleks diferansiyel denklemler, nümerik çözüm 1 Faruk DÜŞÜNCELİ ( ), Mardin Artuklu Üniversitesi, Mimarlık Fakültesi, Mimarlık Bölümü, Mardin, Türkiye 2 Ercan ÇELİK ( ), Atatürk Üniversitesi, Fen Fakültesi, Matematik Bölümü, Erzurum, Türkiye Sorumlu yazar/corresponding Author: Faruk DÜŞÜNCELİ, farukdusunceli@artuklu.edu.tr Geliş tarihi / Received: Kabul tarihi / Accepted:

2 Faruk DÜŞÜNCELİ and Ercan ÇELİK INTRODUCTION Different type of differential equations have been solved with taylor (Sezer and Yalçınbaş, 2009), Bessel (Yüzbaşı et al., 2011), laguerre (Gülsu et al., 2011), hermite (Yüzbaşı et al., 2011), legendre (Tohidi, 2012; Düşünceli and Çelik, 2015) and Fibonacci polynomials (Düşünceli and Çelik, 2017). In this paper, the matrix operates between the Hermite polynomials and their derivatives, we utilized the Hermite method to solve linear complex differential equation. mm nn=0 PP nn (zz)ff (nn) (zz) = gg(zz) (1) with the initial conditions ff (tt) (αα) = θθ tt tt = 0,1,., mm 1 (2) We accept ff(zz) is unknown function,pp nn (zz) and gg(zz) are analytical functions in the circular domain whichdd = {zz = xx + iiii, zz CC, zz rr, rr RR + }; αα DD, θθ tt is appropriate complex or real constant. Suppose that the solution of (1) under the initial conditions (2) is approximated ff(zz) = NN nn=0 aa nn HH nn (zz), zz DD (3) which is the Hermite series of the unknown function ff(zz), where all of aa nn are the Hermite coefficients to be determined.hermite polynomials defined by [ nn 2 ] HH nn (zz) = nn! ( 1)mm mm! (nn 2mm)! (2zz)nn 2mm mm=0 Where[ nn ] = nn if nn is even and 2 2 [nn] = nn 1 if nn is odd and we use the collocation points 2 2 zz pppp = rr NN ppeeiiii NN pp, 0 < θθ 2ππ, rr RR +, pp 0,1,, NN (4) MATERIAL AND METHOD 190 Iğdır Üni. Fen Bilimleri Enst. Der. / Iğdır Univ. J. Inst. Sci. & Tech.

3 Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials MATERIAL AND METHOD We can write the desired solution ff(zz) of Equation (3) ff(zz) = HH(zz)AA (5) where HH(zz) = [HH 0 (zz) HH 1 (zz) HH NN (zz)] and AA = [aa 0 aa 1 aa NN ] TT The Hermite polynomialshh nn (zz) can be formed in matrix form as HH(zz) = ZZ(zz)BB TT (6) where ZZ(zz) = [1 zz zz 2 zz NN ] and whether N is odd, BB = [ 0! ( 1)0 0! 0! ! ( 1)0 0! 1! ! ( 1)1 1! 0! ! ( 1)0 0! 2! ! ( 1)1 1! 1! ! ( 1)0 0! 3! ! ( 1)2 2! 0! ! ( 1)1 1! 2! ! ( 1)0 0! 4! nn! ( 1)( nn 1 2 ) ( nn 1 2 )! 1! 21 0 nn! ( 1) (nn 3 2 ) ( nn 3 2 )! 3! 23 0 nn! ( 1) 0 0! nn! 2nn ] NN+1xN+1 Cilt / Volume: 7, Sayı / Issue: 4,

4 Faruk DÜŞÜNCELİ and Ercan ÇELİK whether N is even, BB = 0! ( 1)0 0! 0! ! ( 1)0 0! 1! ! ( 1)1 1! 0! ! ( 1)0 0! 2! ! ( 1)1 1! 1! ! ( 1)0 0! 3! ! ( 1)2 2! 0! ! ( 1)1 1! 2! nn 2 ) nn! ( 1)( [ ( nn )! 0! 20 0 nn! ( 1) 2 ( nn 2 2 ) ( nn 2 2 )! 2! 22 0 nn! ( 1) 0 0! nn! 2nn 0 ] NN+1xN+1 Then, the relation between the matrix HH(zz) and its derivativeshh (zz), HH (2) (zz),, HH (nn) (zz)are HH (zz) = HH(zz)KK TT HH (2) (zz) = HH(zz)(KK TT ) 2 HH (nn) (zz) = HH(zz)(KK TT ) nn (7) where KK = NN [ ] NN+1xN+1 By using the relations (6) and (7) we obtain the relation ff (nn) (zz) = HH (nn) (zz)kk TT AA = HH(zz)(KK TT ) nn AA = ZZ(zz)BB TT (KK TT ) nn AA (8) By changing the collocation points zz = zz pppp into the relation (8), we get the following matrix equations 192 Iğdır Üni. Fen Bilimleri Enst. Der. / Iğdır Univ. J. Inst. Sci. & Tech.

5 Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials ff (nn) (zz pppp ) = ZZ(zz pppp )BB TT (KK TT ) nn AA, pp 0,1,, NN (9) For pp = 0,1,..., NN, we can write the relation (9) ff (nn) (zz 00 ) = ZZ(zz 00 )BB TT (KK TT ) nn AA ff (nn) (zz 11 ) = ZZ(zz 11 )BB TT (KK TT ) nn AA ff (nn) (zz NNNN ) = ZZ(zz NNNN )BB TT (KK TT ) nn AA where ZZ 00 ZZ 0 (zz 00 ) ZZ 1 (zz 00 ) ZZ 2 (zz 00 ) ZZ NN (zz 00 ) ZZ ZZ = [ 11 ZZ ] = [ 0 (zz 11 ) ZZ 1 (zz 11 ) ZZ 2 (zz 11 ) ZZ NN (zz 11 ) ] ZZ NNNN ZZ 0 (zz NNNN ) ZZ 1 (zz NNNN ) ZZ 2 (zz NNNN ) ZZ NN (zz NNNN ) Let us modify the collocation points (4) into equation(1), mm nn=0 PP nn (zz pppp )ff (nn) (zz pppp )AA = gg(zz pppp ) (10) We attain the basic matrix equation of the relations (8) (10), mm NN PP nn (zz pppp )ZZ(zz pppp )BB TT (KK TT ) nn A = NN nn=0 pp=0 pp=0 GG pp (11) where PP nn (zz 00 ) 0 0 gg(zz 00 ) 0 PP PP nn = [ nn (zz 11 ) 0 gg(zz ]andgg pp = [ 11 ) ] 0 0 PP nn (zz NNNN ) gg(zz NNNN ) Since the A is unknown and should be determined that the matrix equation (11) could be rewritten in the subsequent form: WWWW = GGor[WW; GG] = [ww pppp ; gg pp ] pp, qq = 0,1,, NN (12) where, mm NN pp=0 WW = nn=0 PP nn (zz pppp )ZZ(zz pppp )BB TT (KK TT ) nn andaa = [aa 0 aa 1 aa NN ] TT We write the matrix shape of the initial conditions (2) by the aid of (8), or real constant. Cilt / Volume: 7, Sayı / Issue: 4,

6 Faruk DÜŞÜNCELİ and Ercan ÇELİK We write the matrix shape of the initial conditions (2) by the aid of (8), ff (tt) (αα) = HH(αα)(KK TT ) tt AA = θθ tt tt = 0,1,., mm 1 In another hand the matrix shape of the initial conditions could be reformed as UU tt AA = θθ tt tt = 0,1,., mm 1 where UU tt = HH(αα)(KK TT ) tt tt = 0,1,., mm 1 the augmented form of these equations are [UU tt ; θθ tt ] = [uu tt0, uu tt1,, uu tttt ; θθ tt ] tt = 0,1,., mm 1 (13) Finally, to find the unknown Hermite coefficients connected to the Finally, to find the unknown Hermite coefficientsconnected to the approximate solution of the problem (1) under the initial conditions (2), we require to replace them rows of (13) by the last m rows of the augmented matrix (12) and hence we have new augmented matrix ww 00 ww 01 ww 0NN ; gg 0 ww 10 ww 11 ww 1NN ; gg 1 ww NN mm 0 ww NN mm 1 ww NN mm NN ; gg NN [WW ; GG ] = uu 00 uu 01 uu 0NN ; θθ 0 uu 10 uu 11 uu 1NN ; θθ 1 [ uu mm 1 0 uu mm 1 1 uu mm 1 NN ; θθ mm 1 ] (14) or the matrix equation WW AA = GG (15) If can rewrite (15) in the form and the Ais uniquely set. The If can rewrite (15) in the form and the Ais uniquely set. The solution is given by the Hermite series are determined. And so on, the m th order linear complex differential equation under the initial conditions has an approximated. We can control the precision of the acquired solutions. Since the Hermite series are numerical solution of (1). So the equation should be approximately efficient for 194 Iğdır Üni. Fen Bilimleri Enst. Der. / Iğdır Univ. J. Inst. Sci. & Tech.

7 Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials zz = zz jj DD, jj = 0, 1, 2,... mm EE(zz jj ) = nn=0 PP nn (zz jj )ff (nn) (zz jj ) gg(zz jj ) 0 (16) or EE(zz jj ) 10 ll jj (ll jj is any positive integer). If mmmmmm10 ll jj = 10 ll is described, then the truncation limit N is put on until the values EE(zz jj ) at eachof the points zz jj becomes smaller than the prescribed 10 ll. RESULTS AND DISCUSSION In this part, two examples are given to illustrate the accuracy and effectiveness of the proposed way and all of them are complemented on a computer by using programs typed in Matlab. Therefore, we have recorded in tables, the values of the exact solution ff(xx, yy) = xx(573/ (21ii)/ ) + yy( 21/ (573ii)/ ) + (xx + yyyy) 2 ( / (53ii)/ ) + (xx + yyyy) 3 ( 159/ (7ii)/ ) + (xx + yyyy) 4 ( / ( ii)/ ) + (xx + yyyy) 5 ( / ( ii)/ ) / (19ii)/ For N=10, Cilt / Volume: 7, Sayı / Issue: 4,

8 Faruk DÜŞÜNCELİ and Ercan ÇELİK ff(xx, yy) = xx( / ( ii)/ ) + yy( / ( ii)/ ) + (xx + yyyy) 2 ( / ( ii)/ ) + (xx + yyyy) 3 ( / (735701ii)/ ) + (xx + yyyy) 4 ( / ( ii)/ ) + (xx + yyyy) 5 ( / ( ii)/ ) + (xx + yyyy) 6 ( / ( ii)/ ) + (xx + yyyy) 7 ( / ( ii)/ ) + (xx + yyyy) 8 ( / ( ii)/ ) + (xx + yyyy) 9 ( / ( ii)/ ) + (xx + yyyy) 10 ( / ( ii)/ ) / ( ii)/ Iğdır Üni. Fen Bilimleri Enst. Der. / Iğdır Univ. J. Inst. Sci. & Tech.

9 Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials The solutions of the linear complex differential equation for N = 3,5 and 10 are obtained. The absolute errors shown in Tables 1,2 and in Figures 1,2. Table 1 Comparison of real parts of the exact solution and numerical one s for some values zz jj Exact solution(real) N=3 N=5 N=10 -.9(1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) Table 2 Comparison of imaginer parts of the exact solution and numerical one s for some values zz jj Exact solution(im.) N=3 N=5 N=10 -.9(1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) (1+i) Cilt / Volume: 7, Sayı / Issue: 4,

10 Faruk DÜŞÜNCELİ and Ercan ÇELİK Figure 1. The real parts of the absolute errors functions Figure 2. The imaginer parts of the absolute errors functions Example 2:Finally, consider the linear complex differential equation 198 Iğdır Üni. Fen Bilimleri Enst. Der. / Iğdır Univ. J. Inst. Sci. & Tech.

11 Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials ff (zz) + 2ff (zz) + zzzz(zz) = z 3 + ee zz (zz + 3) + 6zz + 2 with initial conditions ff(0) = 3, ff (0) = 1. The exact solution is ff(zz) = zz 2 + ee zz (zz + 3) + 6zz + 2. The next step of our method, we obtain the numerical solution for N = 3,5,10. The values of the numerical solution in the issue of N = 3,5,10 for both parts of real and imaginary together with the exact solution and absolute errors are supported in figures 3,4,5,6 as follows. Figure 3. The real parts of the exact solution and numerical one s Cilt / Volume: 7, Sayı / Issue: 4,

12 Faruk DÜŞÜNCELİ and Ercan ÇELİK Figure 4. The imaginer parts of the exact solution and numerical one s Figure 5. The real parts of the absolute errors functions Iğdır Üni. Fen Bilimleri Enst. Der. / Iğdır Univ. J. Inst. Sci. & Tech.

13 Numerical Solution for High-Order Linear Complex Differential Equations By Hermite Polynomials Figure 6. The imaginer parts of the absolute errors functions CONCLUSIONS This way is established on reckoning the coefficients in the Hermite series extension of the solution of a linear complex differential equations providing the circular domain is identified by the functions and by the functions Pn(z) and Gn(z). This way is righteous. It may be concluded that the method is an efficient way to discover numerical solutions for linear complex differential equations. On the other hand, the results are quite reliable and good-agreement with the exact solutions. REFERENCES Düşünceli F, Çelik E, An effective tool: Numerical solutions by Legendre polynomials for high-order linear complex differential equations. British Journal of Applied Science &Technology, 8(4): Düşünceli F, Çelik E, Fibonacci matrix polynomial method for linear complex differential equations. Asian Journal of Mathematics and ComputerResearch, 15(3): Gülsu M, Gürbüz B, Öztürk Y, Sezer M, 2011.Laguerre polynomial approach for solving linear delay difference equations.applied Mathematics and Computation, 217: Sezer M, Yalçınbaş S, 2009.A collocation method to solve higher order linear complex differential equations in rectangular domains.numerical Methods for Partial Differential Equations, 26: Tohidi E, Legendre approximation for solving linear HPDEs and comparison with Taylor and Bernoulli matrix methods. Applied Mathematics, 3: Yüzbaşı S, Aynıgül M, Sezer M, 2011.A collocation method using Hermite polynomials for approximate solution of pantograph equations.journal of the Franklin Institute, 348: Yüzbası S, Şahin N, Gülsu M, A collocation approach for solving a class of complex differential equations in elliptic domains.journal of Numerical Mathematics, 19: Cilt / Volume: 7, Sayı / Issue: 4,