In this paper, least squares approximation method is developed for solving a class of linear fractional integro-differential equations comprising Volterra and Fredhlom cases. This method is based on a polynomial of degree n to compute an approximate solution of these equations. The convergence analysis of the proposed method is proved. In addition, to show the accuracy and the efficiency of the proposed method, some examples are presented

D. Jabari Sabeg

K. Maleknejad

R. Ezzati

International Journal of Analysis and Applications

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Approximate Solution of Fractional Integro-Differential Equations by Least Squares Method

International Journal of Analysis and ApplicationsVolume 17, Number 2 (2019), 303-310URL: https://doi.org/10.28924/2291-8639DOI: 10.28924/2291-8639-17-2019-303APPROXIMATE SOLUTION OF FRACTIONAL INTEGRO-DIFFERENTIALEQUATIONS BY LEAST SQUARES METHODD. JABARI SABEG1,∗, R. EZZATI2 AND K. MALEKNEJAD21Department of Mathematics, Bonab Branch, Islamic Azad University, Bonab, Iran2Department of Mathematics, Karaj Branch, Islamic Azad University, Karaj, Iran∗Corresponding author: davood.jabari@bonabiau.ac.irAbstract. In this paper, least squares approximation method is developed for solving a class of linearfractional integro-differential equations comprising Volterra and Fredhlom cases. This method is based ona polynomial of degree n to compute an approximate solution of these equations. The convergence analysisof the proposed method is proved. In addition, to show the accuracy and the efficiency of the proposedmethod, some examples are presented.1. IntroductionFractional calculus is a significant branch of mathematics that is used in many fields of science andengineering [2–4]. Many researchers have investigated the analytic results on the existence and uniqueness ofsolutions of the fractional differential equations [5–8]. As we know, for most fractional differential equations,there are not method to obtain analytic solutions, so numerical techniques must be used. During the pastyears, methods for solving fractional differential equations are developed. Additionally, some methods haverecently been emerged, such as the Adomian decomposition method [10,11], the operational matrix [12,13],the collocation method [14,16], etc.Received 2018-10-06; accepted 2018-11-27; published 2019-03-01.2010 Mathematics Subject Classification. 26A33, 45A05.Key words and phrases. fractional differential equation; least squares; approximation method; convergence analysis.c©2019 Authors retain the copyrightsof their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License.303Int. J. Anal. Appl. 17 (2) (2019) 304In this paper, by using least squares approximation, a numerical method has been shown to linear fractionalintegro-differential equations in the following formDαy(t) = p(t)y(t) + f(t) + λ1∫ 10k1(t, x)y(x) dx+ λ2∫ t0k2(t, x)y(x) dx t ∈ I = [0, 1] (1.1)with the initial conditionsy(k)(0) = dk i = 0, ..,m− 1, m− 1 < α ≤ m, (1.2)where yk(t) stands for the kth-order derivative of y(t) and Dα denotes the Riemann-Liouville fractionalderivative of order α. Clearly, when λ1 = 0, λ2 = 0, the above equation reduces to a linear fractionaldifferential equation.The rest of the paper is organized as follows: In section 2, we will briefly review some notations anddefinitions of the fractional calculus theory are used in the paper. In Section 3, we introduce the leastsquares approximation method for solving Eq. (1.1), and discuss its convergence. In Section 4, we show theefficiency of the proposed method with some numerical examples. Section 5, as the final section, presents aconclusion.2. Brief review of fractional calculusIn this section, notations and definitions of the fractional calculus theory, which are going to be used inthis paper, are presented [1].2.1. Definition. The Riemann-Liouville fractional integral operator Iα of order α ≥ 0 of a function f(x),is defined asIαf(x) =1Γ(α)∫ x0(x− t)(α−1)f(t) dt , α > 0, (2.1)where x > 0 and Γ(.) is the Euler gamma function.The Riemann-Liouville fractional derivative of order α will be denoted by Dα and defined byDαf(x) =dmdxm(Im−αf(x)), (2.2)where m − 1 < α ≤ m,m ∈ N and m is the smallest integer order greater than α. We just mention thefollowing propertyDαxβ =Γ(β + 1)Γ(β + 1− α)xβ−α, β > −1. (2.3)Int. J. Anal. Appl. 17 (2) (2019) 3053. Method of solutionIn this section, we apply the least squares approximation method for solving Eq. (1.1). We define thefollowing operatorT (t, y(t)) = Dαy(t)− p(t)y(t)− f(t)− λ1∫ 10k1(t, x)y(x) dx− λ2∫ t0k2(t, x)y(x) dx. (3.1)We construct Taylor-series expansion for the solution y(t) in Eq.(1.1) asy(t) ' yn(t) =n∑r=0y(r)(0)r!tr =n∑r=0drr!tr. (3.2)Substituting (3.2) into ( 3.1), we haveT (t, yn(t)) = Dαyn(t)− p(t)yn(t)− f(t)− λ1∫ 10k1(t, x)yn(x) dx− λ2∫ t0k2(t, x)yn(x) dx= Dα(n∑r=0drr!tr)− p(t)n∑r=0drr!tr − f(t)−n∑r=0drr!λ1∫ 10k1(t, x)xr dx−n∑r=0drr!λ2∫ t0k2(t, x)xr dx=n∑r=0drΓ(r + 1)r!Γ(r − α+ 1)tr−α − p(t)n∑r=0drr!tr − f(t)−n∑r=0drr!λ1∫ 10k1(t, x)xr dx−n∑r=0drr!λ2∫ t0k2(t, x)xr dx=n∑r=0drγr(t)− f(t),whereγr(t) =Γ(r + 1)r!Γ(r − α+ 1)tr−α − p(t) trr!− λ1r!∫ 10k1(t, x)xr dx− λ2r!∫ t0k2(t, x)xr dx.Let Rn(t) := T (t, yn(t))− T (t, y(t)), t ∈ [0, 1].Remark 3.1. If Rn(t) = 0, then y(t) = yn(t); if limn→∞Rn(t) = 0, then limn→∞yn(t) = y(t).Remark 3.2. For any t ∈ [0, 1], if Rn(t) ≡ 0, then yn(t) is an exact solution of Eqs. (1.1) and (1.2); iflimn→∞Rn(t) = 0, then yn(t) converges to the exact solution of Eqs. (1.1) and (1.2).LetJ = J(dm, dm+1, ..., dn) =∫ 10T 2(t, yn(t))dt. (3.3)The problem is to find real constants dm, dm+1, ..., dn such that these constants will minimize J . A necessarycondition for the constants dm, dm+1, ..., dn to minimize J is thatInt. J. Anal. Appl. 17 (2) (2019) 306∂J∂dj= 0,for each j = m,m+ 1, ..., n. By referring (3.3), we get∂J∂dj= 2[n∑r=0dr∫ 10γr(t)γj(t)dt−∫ 10f(t)γj(t)dt] = 0. (3.4)Thus, we haven∑r=mdr∫ 10γr(t)γj(t)dt =∫ 10f(t)γj(t)dt− βj , (3.5)whereβj =m−1∑r=0dr∫ 10γr(t)γj(t)dt (3.6)for each j = m,m+ 1, ..., n.In order to find yn(t), we have to solve (n−m) a system of linear equations while assuming (n−m) unknownsdr. The system (3.5) can be written in the form:Gd = F (3.7)whereG =(γm, γm) (γm, γm+1) . . . (γm, γn)(γm+1, γm) (γm+1, γm+1) . . . (γm+1, γn)...... . . ....(γn, γm) (γn, γm+1) . . . (γn, γn), (3.8)d = [dm, dm+1, ..., dn]T ,andF = [(γm, f)− βm, (γm+1, f)− βm+1, ..., (γn, f)− βn]TDefinition 3.3. If Eq. (3.7) has a unique solution d, then yn(t) =∑nr=1drr! tr is called an optimal squaredapproximation solution of Eqs. (1.1)-(1.2) defined on a set as span{1, t, t2, ...tn}, t ∈ [0, 1].Remark 3.4. If limn→∞∫ 10T 2(t, yn(t))dt = 0, then the optimal squared approximation solution yn(t) con-verges to the exact solution y(t) of Eqs. (1.1) and (1.2).We are interested to know that as n → ∞ the optimal squared approximation solution yn(t) will convergeto the exact solution y(t) of Eqs. (1.1) and (1.2). This conception is proven in Theorem 3.5.Int. J. Anal. Appl. 17 (2) (2019) 307Theorem 3.5. Suppose y(t), t ∈ [0, 1] is an exact solution and yn(t) is an optimal squared approximationsolution of Eqs. (1.1) and (1.2). If ∃pn(t) =n∑r=1drtr such that ∀t ∈ [0, 1], limn→∞ pn(t) = y(t) thenlimn→∞∫ 10T 2(t, yn(t))dt = 0.Proof. The proof is similar to proof of Theorem 3 in [19].4. Illustrative examplesIn this section, we use the presented method in Section 3 for solving two examples.Example 4.1. For first example, consider the fractional integro-differential equationD0.75y(t) +15t2ety(t)−∫ t0xety(x) dx =6t2.25Γ(3.25),y(0) = 0,where the exact solution is given by y(t) = t3.We applied the presented method with n = 3 for solving this example and achieved the correspondingabsolute errors in Table 1.Table 1: Absolute errors for Example 1 for n = 3.t = 0 t = 0.2 t = 0.4 t = 0.6 t = 0.8 t = 1.00 1.314×10−15 1.045×10−15 1.914×10−16 2.475×10−16 6.717×10−16Example 4.2. Consider the equationDαy(t) +7t212y(t)−∫ 10txy(x) dx−∫ t0(x+ t)y(x) dx =2t2−αΓ(3− α)− t4,y(0) = 0,with the exact solution y(t) = t2. By the presented method in section 3 for n = 2 and different values ofα absolute errors are reported in Table 2.Example 4.3. Consider the equation [20]Dαy(t) + y(t) = t4 − 12t3 − 3Γ(4− α)t3−α +24Γ(5− α)t4−α,y(0) = 0, 0 ≤ α ≤ 1whose exact solution is given by y(t) = t4 − 12 t3.Int. J. Anal. Appl. 17 (2) (2019) 308Table 2: Absolute errors for Example 2.t α = 0.2 α = 0.5 α = 0.7 α = 0.9 α = 1.00.0 0 0 0 0 00.1 0 1.734× 10−17 1.387× 10−17 0 00.2 0 3.469× 10−17 2.775× 10−17 0 00.3 0 4.163× 10−17 4.163× 10−17 0 00.4 0 5.551× 10−17 5.551× 10−17 0 00.5 0 5.551× 10−17 5.551× 10−17 0 00.6 0 5.551× 10−17 5.551× 10−17 0 00.7 0 5.551× 10−17 1.110× 10−16 0 00.8 0 0 1.110× 10−16 0 00.9 0 1.110× 10−16 1.110× 10−16 0 01.0 0 0 2.220× 10−16 0 0By taking different values of α, we solved the above problem by means of the presented method. Themaximum absolute error with the presented method and SCT method [20] for n = 4 are compared in Table3.Table 3: Comparison of maximum absolute error for example 3.α n=4 Present method Method of [20]0.01 1.26×10−14 1.2×10−50.1 1.41×10−14 1.3×10−40.5 5.30 ×10−15 7.8 ×10−40.99 1.30 ×10−15 8.6 ×10−41 1.14 ×10−15 8.6 ×10−4Example 4.4. Consider the equationD32 y(t) + y(t) =6Γ(2.5)t1.5 +6Γ(1.5)t0.5 + t3 + t2,y(0) = 0, y′(t) = 0,whose exact solution is given by y(t) = t3 + t2.By applying the technique described in section 3 with m=4, we approximate solution asy(t) =4∑r=0y(r)(0)r!tr =4∑r=0drr!trInt. J. Anal. Appl. 17 (2) (2019) 309Here, by using Eq. (3.7), we obtain1.00901 0.422206 0.1237620.422206 0.19103 0.05867630.123762 0.0586763 0.0186265d2d3d4 =4.551271.99060.599582 (4.1)Finally by solving Eq.(4.14), we getd2 = 2, d3 = 6, d4 = 0.Thus we can writey(t) = d0 + d1t+ d2t22!+ d3t33!+ d4t44!= t3 + t2,which is the exact solution.5. ConclusionIn this paper, we proposed least squares approximation method to solve a class of linear fractionalintegro-differential equations comprising of Fredholm and Volterra cases based on a polynomial of degreen. The numerical experiments show that the proposed method can be suitable method for solving theseequations.References[1] C. A. Monje, Y. Chen, B. M. 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Approximate Solution of Fractional Integro-Differential Equations by Least Squares Method

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