Research Article Numerical Solution of Fractional Integro-Differential Equations by Least Squares Method and Shifted Chebyshev Polynomial

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1 Mathematical Problems i Egieerig Volume 24, Article ID 43965, 5 pages Research Article Numerical Solutio of Fractioal Itegro-Differetial Equatios by Least Squares Method ad Shifted Chebyshev Polyomial D. Sh. Mohammed Mathematics Departmet, Faculty of Sciece, Zagazig Uiversity, Zagazig, Egypt Correspodece should be addressed to D. Sh. Mohammed; doaashokry23@yahoo.com Received April 24; Revised 7 May 24; Accepted 2 May 24; Published 2 Jue 24 Academic Editor: Kim M. Liew Copyright 24 D. Sh. Mohammed. This is a ope access article distributed uder the Creative Commos Attributio Licese, which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly cited. We ivestigate the umerical solutio of liear fractioal itegro-differetial equatios by least squares method with aid of shifted Chebyshev polyomial. Some umerical examples are preseted to illustrate the theoretical results.. Itroductio May problems ca be modeled by fractioal Itegrodifferetial equatios from various scieces ad egieerig applicatios. Furthermore most problems caot be solved aalytically, ad hece fidig good approximate solutios, usig umerical methods, will be very helpful. Recetly, several umerical methods to solve fractioal differetial equatios (FDEs) ad fractioal Itegrodifferetial equatios (FIDEs) have bee give. The authors i [, 2] applied collocatio method for solvig the followig: oliear fractioal Lagevi equatio ivolvig two fractioal orders i differet itervals ad fractioal Fredholm Itegro-differetial equatios. Chebyshev polyomials method is itroduced i [3 5] for solvig multiterm fractioal orders differetial equatios ad oliear Volterra ad Fredholm Itegro-differetial equatios of fractioal order. The authors i [6] applied variatioal iteratio method for solvig fractioal Itegro-differetial equatios with the olocal boudary coditios. Adomia decompositio method is itroduced i [7, 8] for solvig fractioal diffusio equatio ad fractioal Itegro-differetial equatios. Refereces [9, ] used homotopy perturbatio method for solvig oliear Fredholm Itegro-differetial equatios of fractioal order ad system of liear Fredholm fractioal Itegro-differetial equatios. Taylor series method is itroduced i [] for solvig liear itegrofractioal differetial equatios of Volterra type. The authors i [2, 3] givea applicatio of oliear fractioal differetial equatios ad their approximatios ad existece ad uiqueess theorem for fractioal differetial equatios with itegral boudary coditios. I this paper least squares method with aid of shifted Chebyshev polyomial is applied to solvig fractioal Itegro-differetial equatios. Least squares method has bee studiedi[4 8]. I this paper, we are cocered with the umerical solutio of the followig liear fractioal Itegro-differetial D α φ (x) =f(x) + K (x, t) φ (t) dt, x,t, () with the followig supplemetary coditios: φ (i) () =δ i, <α, N, (2) where D α φ(x) idicates the αth Caputo fractioal derivative of φ(x); f(x), K(x, t) are give fuctios, x ad t are real variables varyig i the iterval [, ], ad φ(x) is the ukow fuctio to be determied. 2. Basic Defiitios of Fractioal Derivatives I this sectio some basic defiitios ad properties of fractioal calculus theory which are ecessary for the formulatio of the problem are give.

2 2 Mathematical Problems i Egieerig Defiitio. Arealfuctiof(x), x>,issaidtobeithe space C μ, μ R, if there exists a real umber p>μsuch that f(x) = x p f (x),wheref (x) C[, ). Defiitio 2. Afuctiof(x), x>,issaidtobeithespace C m μ, m N },iff(m) C μ. Defiitio 3. The left sided Riema-Liouville fractioal itegral operator of order α of a fuctio f C μ, μ, is defied as [9] J α f (x) = x Γ (α) f (t) dt, α (x t) α>, x>, (3) J f (x) =f(x). (4) Defiitio 4. Let f C m, m N }.ThetheCaputo fractioal derivative of f(x) is defied as [2 22] D α f (x) = J m α f m (x), m < α m, m N, D m f (x) Dx m, α = m. (5) Hece, we have the followig properties: () J α J ] f=j α+] f, α, ] >, f C μ, μ >, (2) J α x γ = Γ(γ+) Γ(α+γ+) xα+γ, (3) J α D α m f (x) =f(x) x>, k= α>, γ>, x>, f (k) ( + ) xk k!, m <α m, (4) D α J α f (x) =f(x), x>, m <α m, (5) D α C=, Cis a costat,, β N, β < [α], (6) D α x β = Γ(β+) Γ(β α+) xβ α, β N, β [α], where [α] deoted the smallest iteger greater tha or equal to α ad N =,, 2,...}. 3. Solutio of Liear Fractioal Itegro-Differetial Equatio I this sectio the least squares method with aid of shifted Chebyshev polyomial is applied to study the umerical solutio of the fractioal Itegro-differetial (). Thismethodisbasedoapproximatigtheukow fuctio φ(x) as φ (x) i= (6) (x), x, (7) where T i (x) is shifted Chebyshev polyomial of the first kid which is defied i terms of the Chebyshev polyomial T (x) by the followig relatio [23]: T (x) =T (2x ), (8) ad the followig recurrece formulae: T (x) =2(2x ) T (x) T 2 (x), =2,3,..., (9) with iitial coditios T (x) =, T (x) =2x, () a i, i =,, 2,..., are costats. Substitutig (7)ito()weobtai D α ( i= (x)) =f(x) + Hece the residual equatio is defied as R(x,a,a,...,a ) = i= (x) f(x) i= i= (t)]dt. () (t)]dt. (2) Let S (a,a,...,a ) = [R (x, a,a,...,a )] 2 w (x) dx, (3) where w(x) is the positive weight fuctio defied o the iterval [, ]. Ithisworkwetakew(x) = for simplicity. Thus S(a,a,...,a ) = i= (x) f(x) i= 2 (t)]dt} dx. (4) So, fidig the values of a i, i =,,...,, which miimize S is equivalet to fidig the best approximatio for the solutio of the fractioal Itegro-differetial equatio (). The miimum value of S is obtaied by settig S =, j=,,...,. (5) a j Applyig (5)to(4)weobtai i= (x) f(x) i= (t)]dt} D α T j (x) K (x, t) T j (t) dt} dx. (6)

3 Mathematical Problems i Egieerig 3 By evaluatig the above equatio for j =,,..., we ca obtai a system of ( + ) liear equatios with ( + ) ukow coefficiets a i s.thissystemcabeformedby usig matrices form as follows: A = ( ( ( ( R(x,a )h dx B= ( ( where ( R(x,a )h dx. R(x,a )h dx f (x) h dx R(x,a )h dx... R(x,a )h dx R(x,a )h dx... R(x,a )h dx ),. d. f (x) h dx ),. f (x) h dx ) R(x,a )h dx... R(x,a )h dx ) (7) Colum Row Figure : The matrix iverse of Example. x h j =D α T j (x) K (x, t) T j (t) dt, j=,,...,,.5 R(x,a i )= i= (x) i= (t)]dt,. i=,,...,. (8) Bysolvigtheabovesystemweobtaithevaluesofthe ukowcoefficietsadtheapproximatesolutioof(). 4. Numerical Examples I this sectio, some umerical examples of liear fractioal Itegro-differetial equatios are preseted to illustrate the above results. All results are obtaied by usig Maple 5. Example. Cosider the followig fractioal Itegrodifferetial Approximate Exact Figure 2: Numerical results of Example. D /2 φ (x) = (8/3) x3/2 2x /2 π + x 2 + xtφ (t) dt, x,t, (9) subject to φ() = with the exact solutio φ(x) = x 2 x. Applyig the least squares method with aid of shifted Chebyshev polyomial of the first kid T i (x), i =,,..., at =5, to the fractioal Itegro-differetial equatio (9) we obtai a system of (6) liear equatios with (6) ukow coefficiets a i, i =,,...,5.Thissystem ca be trasformed ito a matrix equatio ad by solvig this matrix equatio we obtai the iverse which is give i Figure ad we obtai the values of the coefficiets. Substitutig the values of the coefficiets ito (7) weobtai the approximate solutio which is the same as the exact solutio ad the results are show i Figure 2.

4 4 Mathematical Problems i Egieerig Colum Figure 3: The matrix iverse of Example 2. Row 5 2 Colum Row Figure 5: The matrix iverse of Example Similarly as i Example applyig the least squares method with aid of shifted Chebyshev polyomial of the first kid T i (x), i=,,..., at =5,tothefractioalItegrodifferetial equatio (2) the umerical results are show i Figures 3 ad 4 adweobtaitheapproximatesolutiowhich isthesameastheexactsolutio. Example 3. Cosider the followig fractioal Itegrodifferetial Approximate Exact x D 5/3 φ (x) = 3 3Γ (2/3) x /3 π 5 x2 4 x + (xt + x 2 t 2 )φ(t) dt, x,t, (22) Figure 4: Numerical results of Example 2. Example 2. Cosider the followig fractioal Itegrodifferetial D 5/6 φ (x) =f(x) + xe t φ (t) dt, x, t, (2) subject to φ() =,where f (x) = 3 x /6 Γ (5/6) ( x 2 ) + (5 2e) x (2) 9 π with the exact solutio φ(x) = x x 3. subject to φ() = φ() = with the exact solutio φ(x) = x 2. Similarly as i Examples ad 2 applyig the least squares method with aid of shifted Chebyshev polyomial of the first kid T i (x), i=,,..., at =5,tothefractioalItegrodifferetial equatio (22) the umerical results are show i Figures 5 ad 6 adweobtaitheapproximatesolutiowhich isthesameastheexactsolutio. 5. Coclusio I this paper we study the umerical solutio of three examples by usig least squares method with aid of shifted Chebyshev polyomial which derives a good approximatio. We show that this method is effective ad has high covergecy rate.

5 Mathematical Problems i Egieerig Approximate Exact.4 x.6.8 Figure 6: Numerical results of Example 3. Coflict of Iterests The author declares that there is o coflict of iterests regardig the publicatio of this paper. Refereces [] A. H. Bhrawy ad M. A. Alghamdi, A shifted Jacobi-Gauss- Lobatto collocatio method for solvig oliear fractioal Lagevi equatio ivolvig two fractioal orders i differet itervals, Boudary Value Problems, vol.22,article62,3 pages, 22. [2] Y. Yag, Y. Che, ad Y. Huag, Spectral-collocatio method for fractioal Fredholm itegro-differetial equatios, Joural of the Korea Mathematical Society, vol.5,o.,pp , 24. [3] A. H. Bhrawy ad A. S. Alofi, The operatioal matrix of fractioal itegratio for shifted Chebyshev polyomials, Applied Mathematics Letters,vol.26,o.,pp.25 3,23. [4] E. H. Doha, A. H. Bhrawy, ad S. S. Ezz-Eldie, Efficiet Chebyshev spectral methods for solvig multi-term fractioal orders differetial equatios, Applied Mathematical Modellig, vol.35,o.2,pp ,2. [5] S. Iradoust-pakchi, H. Kheiri, ad S. Abdi-mazraeh, Chebyshev cardial fuctios: a effective tool for solvig oliear Volterra ad Fredholm itegro-differetial equatios of fractioal order, Iraia Joural of Sciece ad Techology Trasactio A: Sciece,vol.37,o.,pp.53 62,23. [6] S. Iradoust-pakchi ad S. Abdi-Mazraeh, Exact solutios for some of the fractioal itegro-differetial equatios with the olocal boudary coditios by usig the modifcatio of He s variatioal iteratio method, Iteratioal Joural of Advaced Mathematical Scieces,vol.,o.3,pp.39 44,23. [7] S. Saha Ray, Aalytical solutio for the space fractioal diffusio equatio by two-step Adomia decompositio method, Commuicatios i Noliear Sciece ad Numerical Simulatio,vol.4,o.4,pp ,29. [8] R. C. Mittal ad R. Nigam, Solutio of fractioal itegrodifferetial equatios by Adomai decompositio method, Iteratioal Joural of Applied Mathematics ad Mechaics, vol. 4, o. 2, pp , 28. [9] H. Saeedi ad F. Samimi;, He s homotopy perturbatio method for oliear ferdholm itegro-differetial equatios of fractioal order, Iteratioal Joural of Egieerig Research ad Applicatios,vol.2,o.5,pp.52 56,22. [] R. K. Saeed ad H. M. Sdeq, Solvig a system of liear fredholm fractioal itegro-differetial equatios usig homotopy perturbatio method, Australia Joural of Basic ad Applied Scieces,vol.4,o.4,pp ,2. [] S. Ahmed ad S. A. H. Salh, Geeralized Taylor matrix method for solvig liear itegro-fractioal differetial equatios of Volterra type, Applied Mathematical Scieces,vol.5,o.33-36, pp ,2. [2] J. H. He, Some applicatios of oliear fractioal differetial equatios ad their approxi-matios, Bulleti of Sciece, Techology & Society,vol.5,o.2,pp.86 9,999. [3] S. A. Murad, H. J. Zekri, ad S. Hadid, Existece ad uiqueess theorem of fractioal mixed volterra-fredholm itegrodifferetial equatio with itegral boudary coditios, Iteratioal Joural of Differetial Equatios, vol. 2, Article ID 3457, 5 pages, 2. [4] A. J. Jerri, Itroductio to Itegral Equatios with Applicatios, Joh Wiley & Sos, Lodo, UK, 999. [5]S.N.Shehab,H.A.Ali,adH.M.Yasee, Leastsquares method for solvig itegral equatios with multiple time lags, Egieerig & Techology Joural,vol.28,o.,pp , 2. [6] H. Laeli Dastjerdi ad F. M. Maalek Ghaii, Numerical solutio of Volterra-Fredholm itegral equatios by movig least square method ad Chebyshev polyomials, Applied Mathematical Modellig,vol.36,o.7,pp ,22. [7] M. G. Armetao ad R. G. Durá, Error estimates for movig least square approximatios, Applied Numerical Mathematics, vol.37,o.3,pp ,2. [8] C. Zuppa, Error estimates for movig least square approximatios, Bulleti of the Brazilia Mathematical Society,vol.34,o. 2,pp ,23. [9] K. S. Miller ad B. Ross, A Itroductio to the Fractioal Calculus ad Fractioal Differetial Equatios, A Wiley-Itersciece Publicatio, Joh Wiley & Sos, New York, NY, USA, 993. [2] A. Arikoglu ad I. Ozkol, Solutio of fractioal itegrodifferetial equatios by usig fractioal differetial trasform method, Chaos, Solitos & Fractals,vol.4,o.2,pp , 29. [2] F. Maiardi, Fractioal calculus: some basic problems i cotiuum ad statistical mechaics, i Fractals ad Fractioal Calculus i Cotiuum Mechaics (Udie, 996), vol.378of CISM Courses ad Lectures, pp , Spriger, Viea, Austria, 997. [22] I. Podluby, Fractioal Differetial Equatios,vol.98ofMathematics i Sciece ad Egieerig, Academic Press, Sa Diego, Calif, USA, 999, A itroductio to fractioal derivatives, fractioal differetial equatios, to methods of their solutio ad some of their applicatios. [23] J. C. Maso ad D. C. Hadscomb, Chebyshev Polyomials, Chapma & Hall/CRC, Boca Rato, Fla, USA, 23.

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