A new operational matrix for solving twodimensional nonlinear integral equations of fractional order

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1 Jabari Sabeg et al. Cogent Mathematics (07) 4: APPLIED & INTERDISCIPLINARY MATHEMATICS RESEARCH ARTICLE A new operational matri for solving twodimensional nonlinear integral equations of fractional order Received: 09 Februar 07 Accepted: 3 June 07 First Published: 7 June 07 *Corresponding author: R. Ezzati Department of Mathematics Karaj Branch Islamic Azad Universit Karaj Iran ezati@kiau.ac.ir Reviewing editor: Xinguang Zhang Curtin Universit Australia Additional information is available at the end of the article D. Jabari Sabeg R. Ezzati * and K. Maleknejad Abstarct: In this paper first we derive the operational matri of two-dimensional orthogonal triangular functions (D-TFs) for two-dimensional fractional integrals. Then we appl this operational matri and properties of Two-dimensional orthogonal triangular functions to reduce two-dimensional fractional integral equations to a sstem of algebraic equations. Finall in order to show the validit and efficienc we present some numerical eamples. Subjects: Science; Mathematics & Statistics; Applied Mathematics; Computer Mathematics Kewords: two-dimensional orthogonal triangular functions; two-dimensional fractional integral equations; operational matri. Introduction As a branch of mathematics fractional calculus provides an ecellent tool for describing and modeling such comple engineering and scientific phenomena as fluid-dnamic traffic model (He 999) model frequenc-dependent damping behavior of viscoelastic materials (Bagle & Torvik ) economics (Baillie 996) continuum and statistical mechanics (Mainardi 997) solid R. Ezzati ABOUT THE AUTHORS D. Jabbari Sabeg is currentl PhD student in IAU- Karaj Branch Iran. R. Ezzati received his PhD degree in applied mathematics from IAU- Science and Research Branch Tehran Iran in 006. He is an professor in the Department of Mathematics at Islamic Azad Universit Karaj Branch (Iran) fro05. He has published about 0 papers in international journals and he also is the associate editor of Mathematical Sciences (a Springer Open Journal). His current interests include numerical solution of differential and integral equations fuzz mathematics especiall on solution of fuzz sstems fuzz integral equations and fuzz interpolation. K. Maleknejad received his PhD degree in Applied Mathematics in Numerical Analsis area from the Universit of Wales Aberstwth UK in 980. He has been a professor since 00 at IUST. His research interests include numerical in solving ill-posed problems and solving Fredholm and Volterra integral equations. He has authored as the editor-in-chief of the International Journal of Mathematical Sciences which publishers b Springer. PUBLIC INTEREST STATEMENT As we know we can convert man initial and boundar value problems into problems of solving integral equations. So it is important to develop numerical methods for solving integral equations. In this article after deriving the operational matri of two-dimensional orthogonal triangular functions for two-dimensional fractional integrals we reduce two-dimensional fractional integral equations to a sstem of algebraic equations b appling this operational matri. It is necessar to sa that the introduced operational matri in this paper can be applied for solving differential equations of fractional order integro-differential equations of fractional order etc. 07 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. Page of

2 Jabari Sabeg et al. Cogent Mathematics (07) 4: mechanics (Rossikhin & Shitikova 997) and dnamics of interfaces between soft-nanoparticles and rough substrates (Chow 005). Several numerical methods to solve fractional differential equations and fractional integro-differential equations have been recentl presented b man authors. In EI-Wakil Elhanbal and Abdou (006) authors used Adomian decomposition method for Fractional nonlinear differential equations. Saadatmandi and Dehghan in Saadatmandi and Dehghan (00) used the Legendre operational matri to solve fractional-order differential equations. In Saadatmandi (04) Bernstein polnomials were used for solving partial differential equations. In Maleknejad and Asgari (05) used triangular functions for multi-order fractional differential equations. In Chen Liu Turner and Anh (03) two-dimensional fractional percolation equation was solved. In Najafalizadeh and Ezzati (06) we see two-dimensional block pulse operational matri is used for two-dimensional nonlinear integral equations of fractional order. Here we tr to etend the application of D-TFs to solve two-dimensional nonlinear integral equations of fractional order in. Our main aim is to obtain D-TFs operational matri for two-dimensional fractional integral to reduce the original problem to a sstem of algebraic equations. In this paper first we briefl review fractional calculus and one-dimensional triangular functions (D-TFs). In Section 3 we present the approimation of function via D-TFs. Also b using the properties of D-TFs we derive the operational matri of two-dimensional integration of fractional order. Section 4 is devoted to solving two-dimensional nonlinear fractional integral equations b appling the operational matri of integration of fractional order introduced in previous section. In Section 5 we show the accurac and the efficienc of the proposed method through several eamples. Finall a conclusion is given in Section 6.. Brief review for fractional calculus The most commonl used definitions for fractional derivative and fractional integration are Caputo and Riemann Liouville definitions respectivel. Definition. The Riemann Liouville fractional integral operator I α of order α 0 is defined b: I α f () = ( t) (α ) f (t) dt α>0 Γ(α 0 where > 0 and Γ(.) is the Euler gamma function (Monje Chen Vinagre Xue & Feliu 00). (.) The Riemann Liouville integral satisf the following properties: I α I β f () =I α+β f () I α β Γ(β + ) = Γ(α + + β) α+β. (.) Definition. The left-sided mied Riemann Liouville integral of order α of f is defined as Abbas and Benchohra (04): I α f ( ) = θ Γ(r where α =(r r ) (0 ) (0 ) and θ =(0 0). ( s) r ( t) r f (s t) dtds (.3) Some properties of the left-sided mied Riemann Liouville integral are the following: I θ θf ( ) =f ( ) if p g ( ) then I α θ p q = Γ(p + )Γ(q + ) Γ(p + + r )Γ(q + + r ) p+r q+r. (.4) Page of

3 Jabari Sabeg et al. Cogent Mathematics (07) 4: One-dimensional triangular functions Triangular functions are among orthogonal functions that are introduced b authors of Deb Dasgupta and Sarkar (006) Deb Sarkar and Dasgupta (007). Maleknejad and Asgari (05) applied these functions to solve nonlinear integro-differential equations of fractional order. In Deb et al. (006) an m-set of D-TFs over interval [0 T) are defined as: T i (t) = T i (t) = t ih ih t < (i + )h h t ih ih t < (i + )h h where i = 0 m. And h = T m. (.5) Clearl we can define m-set of D-TF vectors as the following: T(t) =[T 0 (t) T (t) T m (t)] T T(t) =[T 0 (t) T (t) T m (t)] T and T(t) =[T(t) T(t)] T. The vector T(t) is called D-TFs vector. The operational matri for fractional integration can be obtained as Maleknejad and Asgari (05): I α T(t) =p α T(t)+pα T(t) I α T(t) =p α 3 T(t)+pα 4 T(t) (.6) (.7) where p α = p α 3 = and ξ r = ζr = 0 ξ ξ ξ m 0 0 ξ ξ m ξ m ζ ζ ζ m 0 0 ζ ζ m ζ m p α = p α = 4 h α Γ(α + ) ((α + )rα r α+ +(r ) α+ ) h α Γ(α + ) (rα+ (r ) α+ (α + )(r ) α ) ξ ξ ξ 3 ξ m 0 ξ ξ ξ m 0 0 ξ ξ m ξ ζ ζ ζ 3 ζ m 0 ζ ζ ζ m 0 0 ξ ξ m ζ Page 3 of

4 Jabari Sabeg et al. Cogent Mathematics (07) 4: so I α T(t) =P α T(t) where P α fractional integration operational matri of T(t) is ( p α p α p α 3 p α 4 ). (.8) 3. Two-dimensional triangular functions (D-TFs) In Babolian Maleknejad Roodaki and Almasieh (00) authors defined an m -set of D-TFs on [0 T ) [0 T ) as follows: T (s t) = T (s t) = T (s t) = s ih ( )( t jh ) ih h h s < (i + )h jh t < (j + )h T (s t) = s ih ( )( t jh ) ih h h s < (i + )h jh t < (j + )h s ih ( )( t jh ) ih h h s < (i + )h jh t < (j + )h ( s ih h )( t jh h ) ih s < (i + )h jh t < (j + )h where i = 0 m j = 0 and h = T h = T. m m and are arbitrar positive integers. Also the defined the following vectors: T(s t) =[T 00 (s t) T 0 (s t) T (s m t)]t T(s t) =[T 00 (s t) T 0 (s t) T (s m t)]t T(s t) =[T 00 (s t) T 0 (s t) T (s m t)]t T(s t) =[T 00 (s t) T 0 (s t) T (s m t)]t. (3.) With the following properties: T (s t) =T i (s).t j (t) T (s t) =T i (s).t j (t) T (s t) =T i (s).t j (t) T (s t) =T i (s).t j (t). B considering above vectors authors of Babolian et al. (00) defined D-TF vector as the following form: (3.) T(s t) =[T(s t) T(s t) T(s t) T(s t)] T. (3.3) According to this fact to construct the operational matri of D-TFs for the fractional integration in Section 4 we need to derive T(s t) T(s t) T(s t) and T(s t) b Kronecker product of T(t) and T(s). So using (3.) and (3.) we can write T(s t) =T(s) T(t) T(s t) =T(s) T(t) T(s t) =T(s) T(t) T(s t) =T(s) T(t) (3.4) Page 4 of

5 Jabari Sabeg et al. Cogent Mathematics (07) 4: where denotes the Kronecker product defined for two arbitrar matrices A and B as A B =(a ij B) and also it has the following two basic properties (Zhang & Ding 03): (A B)(C D) =(AC) (BD) (A + B) C = A C + B C. In Babolian et al. (00) it is proved that D-TFs are disjoint orthogonal. Thus for ever (4m 4m )-matri B we can write (3.5) T T (s t).b.t(s t) B.T(s t) (3.6) where B is a 4m -vector with elements equal to the diagonal entries of matri B. Also T(s t).t T (s t).x X.T(s t) (3.7) where X is a 4m -vector and X = diag(x) Regarding to orthogonalit of D-TFs a function f(s t) defined over ([0 T ) [0 T )) can be epanded b D-TFs as Babolian et al. (00): f (s t) m i=0 m i=0 j=0 j=0 c T (s t)+ e T (s t)+ m i=0 m i=0 j=0 j=0 d T (s t)+ l T (s t) = C T T(s t)+d T T(s t)+e T T(s t)+l T T(s t) = F T T(s t) (3.8) where F is a 4m -vector given b: F =[C T D T E T L T ] T and c = f (ih jh ) d = f (ih (j + )h ) e = f ((i + )h jh ) l = f ((i + )h (j + )h ) (3.9) the vector F is called the D-TFs coefficient vector. Authors of Babolian et al. (00) prove that [f (s t)] p F T pt(s t) (3.0) where F p is a column vector whose elements are pth powers of the elements of the vector F and p is the positive integer. Also for a function k( s t) defined on ([0 T ) [0 T ) [0 T 3 ) [0 T 4 )) we have Babolian et al. (00): Page 5 of

6 Jabari Sabeg et al. Cogent Mathematics (07) 4: k( s t) T T ( )KT(s t) (3.) where T( ) and T(s t) are D-TFs vectors of dimension 4m and 4m 3 m 4 respectivel and K is a (4m 4m 3 m 4 ) D-TFs coefficient matri. 4. Operational matri of D-TFs for the fractional integration In this section we construct operational matri of D-TFs for the fractional integration. Using Equations (.3) (3.3) we have: Γ(r = ( s) r ( t) r T(s t) dtds Γ(r 0 0 ( s)r ( t) r T(s t) dtds Γ(r 0 0 ( s)r ( t) r T(s t) dtds Γ(r 0 0 ( s)r ( t) r T(s t) dtds Γ(r 0 0 ( s)r ( t) r T(s t) dtds Using Equation (3.4) we conclude that Γ(r 0 0 ( s)r ( t) r T(s) T(t)dtds Γ(r 0 0 ( s)r ( t) r T(s) T(t)dtds Γ(r 0 0 ( s)r ( t) r T(s) T(t)dtds Γ(r 0 0 ( s)r ( t) r T(s) T(t)dtds = ( Γ(r ) 0 s)(r ) T(s) ds ( Γ(r ) 0 t)(r ) T(t) dt ( Γ(r ) 0 s)(r ) T(s) ds ( Γ(r ) 0 t)(r ) T(t) dt ( Γ(r ) 0 s)(r ) T(s) ds ( Γ(r ) 0 t)(r ) T(t) dt ( Γ(r ) 0 s)(r ) T(s) ds ( Γ(r ) 0 t)(r ) T(t) dt and also b appling Equations (.6) (.7) we obtain (p r T()+pr T()) ((pr T()+pr (p r T()+pr T()) ((pr 3 T()+pr (p r 3 T()+pr 4 T()) ((pr T()+pr (p r 3 T()+pr 4 T()) ((pr 3 T()+pr B appling Equation (3.5) we get T()) T()) 4 T()) = T()) 4 (p r T() pr T())+(pr T() pr T())+(pr T() pr T()) + (pr T() pr (p r T() pr 3 T())+(pr T() pr 4 T())+(pr T() pr 3 T()) + (pr T() pr (p r 3 T() pr T())+(pr 3 T() pr T())+(pr 4 T() pr T()) + (pr 4 T() pr (p r 3 T() pr 3 T())+(pr 3 T() pr 4 T())+(pr 4 T() pr 3 T()) + (pr 4 T() pr T()) T()) 4 T()) T()) 4 = = (p r pr )(T() T()) + (pr pr )(T() T())+(pr pr )(T() T()) + (pr (p r pr 3 )(T() T()) + (pr pr 4 )(T() T())+(pr pr 3 )(T() T()) + (pr (p r )(T() T()) + (pr )(T() T())+(pr 4 pr )(T() T()) + (pr (p r 3 )(T() T()) + (pr 4 )(T() T())+(pr 4 pr 3 )(T() T()) + (pr pr pr 4 4 pr 4 pr 4 )(T() T()) )(T() T()) )(T() T()) )(T() T()) Now b emploing Equation (3.4) we have Page 6 of

7 Jabari Sabeg et al. Cogent Mathematics (07) 4: (p r pr )T( )+(pr pr )T( )+(pr pr )T( )+(pr (p r pr 3 )T( )+(pr pr 4 )T(s t)+(pr pr 3 )T( )+(pr (p r )T( )+(pr )T( )+(pr 4 pr )T( )+(pr (p r 3 )T( )+(pr 4 )T( )+(pr 4 pr 3 )T( )+(pr pr pr 4 4 pr 4 pr 4 )T( ) )T( ) )T( ) )T( ) = p r r where P r r 4m 4m operational matri of fractional integration of T() is p r r = Hence T( ) T( ) T( ) T( ) p r pr p r pr p r pr p r pr p r pr p r 3 pr p r 4 pr p r 3 pr 4 p r 3 pr p r 3 pr p r 4 pr p r 4 pr p r 3 pr p r 3 3 pr p r 4 4 pr p r 3 4 pr 4. Γ(r ( s) r ( t) r T(s t) dtds = p r r 5. Numerical solution of two-dimensional nonlinear fractional integral equations In this section we present an effective method to solve two-dimensional nonlinear integral equations of fractional order. For this purpose we appl two-dimensional triangular functions to approimate known and unknown functions whose properties of these functions were shown in Section 3. Consider the following two-dimensional nonlinear fractional integral equation f ( ) Γ(r where r > 0 r > 0 the functions k( s t) and g( ) are known and f( ) is the unknown function to be determined. Also p is a positive integers. Using the methods mentioned in Section 4 the functions f( ) g( ) [f ( )] p and k( s t) can be approimated b: T( ) T( ) T( ) T( ). ( s) r ( t) r k( s t)[f (s t)] p dtds = g( ) (4.) (5.) f ( ) =T( ) T F g( ) =T( ) T G [f ( )] p = T( ) T F p k( s t) =T( ) T KT(s t) (5.) where T( ) is defined in Equation (3.3) the vectors F G F p and matri K are D-TFs coefficients of f( ) g( ) [f ( )] p and k( s t) respectivel. Now b substituting Equation (5.) in Equation (5.) we have T T ( )F Γ(r Using Equation (3.7) we conclude that ( s) r ( t) r T T ( )KT(s t)t T (s t)f p dtds = T T ( )G. Page 7 of

8 Jabari Sabeg et al. Cogent Mathematics (07) 4: T T ( )F TT ( )KF p Γ(r ( s) r ( t) r T(s t) dtds = T T ( )G. (5.3) Substituting Equation (4.) in Equation (5.3) we have T T ( )F T T ( )K F p P r r T( ) =TT ( )G (5.4) Clearl b assuming H = K F p P r r we will get T T ( )F T T ( )HT( ) =T T ( )G. Now using (3.7) we have: T T ( )F T T ( ) H = T T ( )G Since F p is a diagonal matri we conclude that H = BF p where B is (4m 4m )- matri with components B = K P r r. Hence we will get the following nonlinear algebraic sstem: F BF p = G. (5.5) Clearl this sstem can be solved b known methods such as Newton s method. After solving (5.5) we can obtain the approimate solution of (5.) using (3.8). 6. Illustrative eamples To illustrate the effectiveness of the proposed method in the Section 5 we present three test eamples. In these eamples we assume that T = T = m =. Also in this section we appl the following error function e( ) = f ( ) f m ( ) where f( ) and f m ( ) are the eact and the approimate solutions of the two-dimensional fractional integral Equation (5.) respectivel. Eample 6. (Najafalizadeh & Ezzati 06) Consider the following two-dimensional fractional integral equation: f ( ) Γ( 3 )Γ( 5 ( s) ( t) 3 t[f (s t)] dtds = ( ) 3 whose eact solution is given b f ( ) = 3 3. The approimate solution of f( ) is obtained using D-TFs method described in Section 5. Table shows a comparison of the proposed method and the method of Najafalizadeh and Ezzati (06). The displaed results show that the proposed method is more accurate than the proposed method in Najafalizadeh et al. (06). Page 8 of

9 Jabari Sabeg et al. Cogent Mathematics (07) 4: Table. Numerical results for Eample 6. Absolute error Proposed method Proposed method Method of Najafalizadeh and Ezzati (06) = m = 6 m = 8 m = Ma error Eample 6. Consider the following two-dimensional fractional integral equation: f ( ) Γ( 7 )Γ( 7 ( s) 5 ( t) 5 tf (s t) dtds = The eact solution of this eample is f ( ) =. Table illustrates the numerical results for this eample. Eample 6.3 As a last eample we present the following two-dimensional fractional integral equation: f ( ) Γ( 9 )Γ( 3 ( s) 7 ( t) 5 s( )[f (s t)] dtds ) = ( 6 3 ( )( ) 4 Table. The numerical results for Eample 6. = m = 4 m = 6 m = 8 Eact solution Ma error Page 9 of

10 Jabari Sabeg et al. Cogent Mathematics (07) 4: Table 3. The numerical results for Eample 6.3 = m = 4 m = 6 m = 0 Eact solution Ma error with the eact solution f ( ) =. Table 3 illustrates the numerical results for this eample. 7. Conclusion A general formulation for D-TFs operational matri for two-dimensional fractional integral equations has been derived. This matri is used to approimate the numerical solution of the two-dimensional nonlinear fractional integral equations. The properties of D-TFs and the operational matri are used to reduce the problem to a sstem of algebraic equations that can be solved b known methods. Finall we presented three numerical eamples to demonstrate the validit and applicabilit of the proposed method. Funding The authors received no direct funding for this research. Author details D. Jabari Sabeg davood.jabari@bonabiau.ac.ir R. Ezzati ezati@kiau.ac.ir ORCID ID: K. Maleknejad maleknejad@iust.ac.ir Department of Mathematics Karaj Branch Islamic Azad Universit Karaj Iran. Citation information Cite this article as: A new operational matri for solving two-dimensional nonlinear integral equations of fractional order D. Jabari Sabeg R. Ezzati & K. Maleknejad Cogent Mathematics (07) 4: References Abbas S. & Benchohra M. (04). Fractional order in tegrale quations of two in dependent variables. Applied Mathematics and Computation Babolian E. Maleknejad K. Roodaki M. & Almasieh H. (00). Two-dimensional triangular functions and their applications to nonlinear D Volterra-Fredholm integral equations. Computers & Mathematics with Applications Bagle R. L. & Torvik P. J. (983). A theoretical basis for the application of fractional calculus to viscoelasticit. Journal of Rheolog Bagle R. L. & Torvik P. J. (985). Fractional calculus in the transient analsis of viscoelasticall damped structures. The American Institute of Aeronautics and Astronautics Journal Baillie R. T. (996). Long memor processes and fractional integration in econometrics. Journal of Econometrics Chen S. Liu F. Turner I. & Anh V. (03). An implicit numerical method for the two-dimensional fractional percolation equation. Applied Mathematics and Computation Chow T. S. (005). Fractional dnamics of interfaces between soft-nanoparticles and rough substrates. Phsics Letters A Deb A. Dasgupta A. & Sarkar G. (006). A new set of orthogonal functions and its application to the analsis of dnamic sstems. Journal of The Franklin Institute Deb A. Sarkar G. & Sengupta A. (007). Triangular orthogonal functions for the analsis of continuous time sstems. Elsevier. EI-Wakil S. A. Elhanbal A. & Abdou M. A. (006). Adomian decomposition method for solving fractional nonlinear differential equations. Applied Mathematics and Computation He J. H. (999). Some applications of nonlinear fractional differential equations and their approimations. Bulletin of Science Technolog & Societ Mainardi F. (997). Fractional calculus: Some basic problems in continuum and statistical mechanics. In A. Carpinteri & F. Mainardi (Eds.) Fractals and fractional calculus in continuum mechanics (pp ). Springer Verlag: New York. Maleknejad K. & Asgari M. (05). The construction of operational matri of fractional integration using triangular functions. Applied Mathematical Modelling Page 0 of

11 Jabari Sabeg et al. Cogent Mathematics (07) 4: Monje C. A. Chen Y. Vinagre B. M. Xue D. & Feliu V. (00). Fractional-order sstems and controls. Advances in industrial control. London: Springer. Najafalizadeh S. & Ezzati R. (06). Numerical methods for solving two-dimensional nonlinear integral equations of fractional order b using two-dimensional block pulse operational matri. Applied Mathematics and Computation Rossikhin Y. A. & Shitikova M. V. (997). Applications of fractional calculus to dnamic problems of linear and nonlinear hereditar mechanics of solids. Applied Mechanics Reviews Saadatmandi A. (04). Bernstein operational matri of fractional derivatives and its applications. Applied Mathematical Modelling Saadatmandi A. & Dehghan M. (00). A new operational matri for solving fractional-order differential equations. Journal of Computational and Applied Mathematics Zhang H. & Ding F. (03). On the Kronecker products and their applications. Journal of Applied Mathematics The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4.0 license. You are free to: Share cop and redistribute the material in an medium or format Adapt remi transform and build upon the material for an purpose even commerciall. The licensor cannot revoke these freedoms as long as ou follow the license terms. Under the following terms: Attribution You must give appropriate credit provide a link to the license and indicate if changes were made. You ma do so in an reasonable manner but not in an wa that suggests the licensor endorses ou or our use. No additional restrictions You ma not appl legal terms or technological measures that legall restrict others from doing anthing the license permits. Cogent Mathematics (ISSN: ) is published b Cogent OA part of Talor & Francis Group. Publishing with Cogent OA ensures: Immediate universal access to our article on publication High visibilit and discoverabilit via the Cogent OA website as well as Talor & Francis Online Download and citation statistics for our article Rapid online publication Input from and dialog with epert editors and editorial boards Retention of full copright of our article Guaranteed legac preservation of our article Discounts and waivers for authors in developing regions Submit our manuscript to a Cogent OA journal at Page of