AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 617010 10.1155/2013/617010 617010 Research Article New Iterative Method: An Application for Solving Fractional Physical Differential Equations Hemeda A. A. Chung Soon Y. Department of Mathematics Faculty of Science Tanta University Tanta 31527 Egypt tanta.edu.eg 2013 13 5 2013 2013 12 09 2012 07 03 2013 2013 Copyright © 2013 A. A. Hemeda. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The new iterative method with a powerful algorithm is developed for the solution of linear and nonlinear ordinary and partial differential equations of fractional order as well. The analysis is accompanied by numerical examples where this method, in solving them, is used without linearization or small perturbation which confirm the power, accuracy, and simplicity of the given method compared with some of the other methods.

1. Introduction

Considerable attention has been devoted to the study of the fractional calculus during the past three decades and its numerous applications in the area of physics and engineering. The applications of fractional calculus used in many fields such as electrical networks, control theory of dynamical systems, probability and statistics, electrochemistry, chemical physics, optics, and signal processing can be successfully modelled by linear or nonlinear fractional differential equations.

So far there have been several fundamental works on the fractional derivative and fractional differential equations . These works are to be considered as an introduction to the theory of fractional derivative and fractional differential equations and provide a systematic understanding of the fractional calculus such as the existence and uniqueness [4, 5]. Recently, many other researchers have paid attention to existence result of solution of the initial value problem and boundary problem for fractional differential equations .

Finding approximate or exact solutions of fractional differential equations is an important task. Except for a limited number of these equations, we have difficulty in finding their analytical solutions. Therefore, there have been attempts to develop new methods for obtaining analytical solutions which reasonably approximate the exact solutions. Several such techniques have drawn special attention, such as Adomain’s decomposition method , homotopy perturbation method , homotopy analysis method [11, 12], variational iteration method , Chebyshev spectral method [18, 19], and new iterative method . Among them, the new iterative method provides an effective procedure for explicit and numerical solutions of a wide and general class of differential systems representing real physical problems. The new iterative method is more superior than the other nonlinear methods, such as the perturbation methods where this method does not depend on small parameters, such that it can find wide application in nonlinear problems without linearization or small perturbation.

The motivation of this paper is to extend the application of the new iterative method proposed by Daftardar-Gejji and Jafari  to solve linear and nonlinear ordinary and partial differential equations of fractional order. This motivation is based on the importance of these equations and their applications in various subjects in physical branches [10, 11, 14, 2325].

There are several definitions of a fractional derivative of order α>0 [3, 26]. The two most commonly used definitions are Riemann-Liouville and Caputo. Each definition uses Riemann-Liouville fractional integration and derivative of whole order. The difference between the two definitions is in the order of evaluation. Riemann-Liouville fractional integration of order α is defined as (1)Iαf(x)=1Γ(α)0x(x-ξ)α-1f(ξ)dξ,α>0,x>0. The next two equations define Riemann-Liouville and Caputo fractional derivatives of order α,  respectively, as(2a)Dαf(x)=dmdxm(Im-αf(x)),(2b)D*αf(x)=Im-α(dmdxmf(x)),where m-1<α<m,  mN.

Caputo fractional derivative first computes an ordinary derivative followed by a fractional integral to achieve the desired order of fractional derivative. Riemann-Liouville fractional derivative is computed in the reverse order. Therefore, Caputo fractional derivative allows traditional initial and boundary conditions to be included in the formulation of the problem.

From properties of D*α  and Iα,  it is important to note that(3a)D*αxβ=Γ(β+1)xβ-αΓ(β+1-α),βα,   where D*α  is Caputo derivative operator of order α, (3b)Iαxβ=Γ(β+1)xβ+αΓ(β+1+α).

2. Basic Idea of New Iterative Method

For the basic idea of the new iterative method, we consider the following general functional equation : (4)u(x)=f(x)+N(u(x)), where N is a nonlinear operator from a Banach space BB and f  is a known function. We have been looking for a solution of (4) having the series form (5)u(x)=i=0ui(x). The nonlinear operator N can be decomposed as (6)N(i=0ui)=N(u0)+i=1{N(j=0iuj)-N(j=0i-1uj)}. From (5) and (6), (4) is equivalent to (7)i=0ui=f+N(u0)+i=1{N(j=0iuj)-N(j=0i-1uj)}. We define the following recurrence relation: (8)u0=f,u1=N(u0),un+1=N(u0+u1++un)-N(u0+u1++un-1),n=1,2,. Then, (9)(u1+u2++un+1)=N(u0+u1++un),n=1,2,,u=i=0ui=f+N(i=0ui). If N(x)-N(y)<kx-y,  0<k<1, then (10)un+1=N(u0++un)-N(u0++un-1)kunkn+1u0,n=0,1,2,, and the series i=0ui absolutely and uniformly converges to a solution of (4) , which is unique, in view of the Banach fixed point theorem . The n-term approximate solution of (4) and (5) is given by u(x)=i=0n-1ui.

2.1. Convergence of the Method

Now we analyze the convergence of the new iterative method for solving any general functional equation (4). Let e=u*-u, where u* is the exact solution, u is the approximate solution, and e is the error in the solution of (4); obviously e satisfies (4), that is, (11)e(x)=f(x)+N(e(x)) and the recurrence relation (8) becomes (12)e0=f,e1=N(e0),en+1=N(e0+e1++en)-N(e0+e1++en-1),n=1,2,. If N(x)-N(y)kx-y,  0<k<1, then (13)e0=f,e1=N(e0)ke0,e2=N(e0+e1)-N(e0)ke1k2e0,e3=N(e0+e1+e2)-N(e0+e1)ke2k3e0,en+1=N(e0++en)-N(e0++en-1)kenkn+1e0,n=0,1,2,. Thus en+10 as n, which proves the convergence of the new iterative method for solving the general functional equation (4). For more details, you can see .

3. Suitable Algorithm

In this section, we introduce a suitable algorithm for solving nonlinear partial differential equations using the new iterative method. Consider the following nonlinear partial differential equation of arbitrary order:(14a)D*tαu(x,t)=A(u,u)+B(x,t),m-1<α<m,mN,(14b)ktku(x,0)=hk(x),k=0,1,,m-1,where A is a nonlinear function of u and u (partial derivatives of u with respect to x and t) and B is the source function. In view of the new iterative method, the initial value problem (14a) and (14b) is equivalent to the integral equation (15)u(x,t)=k=0m-1hk(x)tkk!+ItαB+ItαA=f+N(u), where(16a)f=k=0m-1hk(x)tkk!+ItαB,(16b)N(u)=ItαA.

Remark 1.

When the general functional equation (4) is linear, the recurrence relation (8) can be simplified in the form (17)u0=f,un+1=N(un),n=0,1,2,.

Proof.

From the properties of integration and by using (8) and (16b), we have (18)un+1=N(u0++un-1+un)-N(u0++un-1)=Itα[u0++un-1+un]-Itα[u0++un-1]=Itα[u0]++Itα[un-1]+Itα[un]-Itα[u0]-Itα[un-1]=Itα[un]=N(un),n=0,1,2,. Therefore, we get the solution of (15) by employing the recurrence relation (8) or (17).

4. Applications

To illustrate the effectiveness of the proposed method, several test examples are carried out in this section.

Example 2.

In this example, we consider the following initial value problem in the case of the inhomogeneous Bagely-Torvik equation [23, 24]: (19)D*x2u(x)+D*x1.5u(x)+u(x)=g(x),u(0)=1,u(0)=1,x[0,L], where g(x)=1+x. The exact solution of this problem is u(x)=1+x.

By applying the technique described in Sections 2 and 3, the initial value problem (19) is equivalent to the integral equation (20)u(x)=1+x+x22+x36-Ix2[D*x1.5u(x)+u(x)].

Let N(u)=-Ix2[D*x1.5u(x)+u(x)]. In view of recurrence relation (17), we have the following first approximations: (21)u0(x)=1+x+x22+x36,u1(x)=N(u0)=-8x2.515π-16x3.5105π-x22-x36-x424-x5120,u2(x)=N(u1)=8x2.515π+16x3.5105π+64x4.5945π+128x5.510395π+x36+x412+x5120+x6720+x75040, and so on. In the same manner the rest of components can be obtained. The 6-term approximate solution for (19) is (22)u(x)=i=05ui=1+x-x6144-5x75040-x936288-x10362880-x12479001600-x136227020800-32x4.5945π-64x5.510395π-512x7.5405405π-1024x8.56891885π-512x10.5687465529π-128x11.51976463395π.

Remark 3.

In Example 2. we have used the recurrence relation (17). If we used the recurrence relation (8) in place of (17), we obtain the same result.

In Figure 1, we have plotted the 6-term approximate solution with the corresponding exact solution for (19). It is remarkable to note that the two solutions are almost equal.

Plots of the approximate solution and the exact solution for (19).

Comparing these obtained results with those obtained by new Jacobi operational matrix in [23, 24], we can confirm the simplicity and accuracy of the given method.

Example 4.

Consider the following fractional Riccati equation : (23)D*xαu(x)+u2(x)=1,u(0)=0,x>0,0<α1. The exact solution when α=1 is u(x)=(e2x-1)/(e2x+1).

By applying the technique described in Sections 2 and 3, the initial value problem (23) is equivalent to the integral equation (24)u(x)=xαΓ(1+α)-Ixα[u2(x)].

Let N(u)=-Ixα[u2(x)]. In view of recurrence relation (8), we have the following first approximations: (25)u0(x)=xαΓ(1+α),u1(x)=N(u0)=-Γ(1+2α)x3αΓ(1+3α)Γ(1+α)2,u2(x)=N(u0+u1)-N(u0)=2Γ(1+2α)Γ(1+4α)x5αΓ(1+3α)Γ(1+5α)Γ(1+α)3-Γ(1+2α)2Γ(1+6α)x7αΓ(1+3α)2Γ(1+7α)Γ(1+α)4, and so on. The 4-term approximate solution for (23) is (26)u(x)=i=03ui=xαΓ(1+α)-Γ(1+2α)x3αΓ(1+3α)Γ(1+α)2+2Γ(1+2α)Γ(1+4α)x5αΓ(1+3α)Γ(1+5α)Γ(1+α)3-Γ(1+2α)2Γ(1+6α)x7αΓ(1+3α)2Γ(1+7α)Γ(1+α)4-4Γ(1+2α)Γ(1+4α)Γ(1+6α)x7αΓ(1+3α)Γ(1+5α)Γ(1+7α)Γ(1+α)4+4Γ(1+2α)2Γ(1+4α)Γ(1+3α)2Γ(1+5α)·Γ(1+8α)x9αΓ(1+9α)Γ(1+α)5+2Γ(1+2α)2Γ(1+6α)Γ(1+8α)x9αΓ(1+3α)2Γ(1+7α)Γ(1+9α)Γ(1+α)5-4Γ(1+2α)2Γ(1+4α)2Γ(1+10α)x11αΓ(1+3α)2Γ(1+5α)2Γ(1+11α)Γ(1+α)6-2Γ(1+2α)3Γ(1+3α)3Γ(1+7α)·Γ(1+6α)Γ(1+10α)x11αΓ(1+11α)Γ(1+α)6+4Γ(1+2α)3Γ(1+4α)Γ(1+6α)Γ(1+3α)3Γ(1+5α)Γ(1+7α)·Γ(1+12α)x13αΓ(1+13α)Γ(1+α)7-Γ(1+2α)4Γ(1+6α)2Γ(1+14α)x15αΓ(1+3α)4Γ(1+7α)2Γ(1+15α)Γ(1+α)8.

In Figure 2, we have plotted the 4-term approximate solution for (23) for different values of α with the corresponding exact solution. It is remarkable to note that the approximate solution,  in case α=1,  and the exact solution are almost equal (continuous curve) whenever the approximate solution, in cases α=0.9,0.8, is of high agreement with the exact solution (dashed and dotted curves, resp.).

Plots of the approximate solution for different values of α and the exact solution for (23).

Comparing the obtained results with those obtained by homotopy analysis method, in case h=-1, in , we can confirm the simplicity and accuracy of the given method.

Example 5.

Consider the following initial value problem with fractional order [23, 24]: (27)D*x3u(x)+D*x2.5u(x)+u2(x)=x4,u(0)=u(0)=0,u′′(0)=2.

The exact solution for this problem is u(x)=x2.

As in Example 4, the initial value problem (27) is equivalent to the integral equation (28)u(x)=x2+x7210-Ix3[D*x2.5u(x)+u2(x)].

Let N(u)=-Ix3[D*x2.5u(x)+u2(x)]. In view of recurrence relation (8), we have the following first approximations: (29)u0(x)=x2+x7210,u1(x)=N(u0)=-129024x7.542567525π-x7210-x12138600-x17179928000,u2(x)=N(u0+u1)-N(u0)=1.71008-3x7.5+5.95252-4x8+7.21501-6x12+4.28636-6x12.5+9.09455-9x17+1.31911-9x17.5-5.97301-10x18+1.20298-12x22-2.48834-12x22.5-2.96617-15x27-1.02289-15x27.5-2.69485-18x32-6.62568-28x37, and so on. The 4-term approximate solution and the corresponding exact solution for (27) are plotted in Figure 3. It is remarkable to note that the two solutions are almost equal.

Plots of the approximate solution and the exact solution for (27).

Comparing these obtained results with those obtained by new Jacobi operational matrix in [23, 24], we can confirm the simplicity and accuracy of the given method.

Example 6.

Consider the following fractional order wave equation in 2-dimensional space : (30)D*tαu(x,y,t)+c(ux+uy)=0,u(x,y,0)=sin[π(x+yl)],0<α1.

The exact solution for this problem when α=1 is (31)u(x,y,t)=sin[π(x+y-2ctl)].

The initial value problem (30) is equivalent to the integral equation (32)u(x,y,t)=sin[π(x+yl)]-Itα[c(ux+uy)].

Let N(u)=-Itα[c(ux+uy)].  In view of recurrence relation (17), we have the following first approximations: (33)u0(x,y,t)=sin[π(x+yl)],u1(x,y,t)=-1Γ(1+α)(2cπtαl)·cos[π(x+yl)],u2(x,y,t)=-1Γ(1+2α)(2cπtαl)2·sin[π(x+yl)],u3(x,y,t)=1Γ(1+3α)(2cπtαl)3·cos[π(x+yl)], and so on. The n-term approximate solution for (30) is (34)u(x,y,t)=i=0n-1ui=sin[π(x+yl)]×[1Γ(1+6α)(2cπtαl)61-1Γ(1+2α)(2cπtαl)2+1Γ(1+4α)(2cπtαl)4-1Γ(1+6α)(2cπtαl)6+]-cos[π(x+yl)]×[1Γ(1+7α)(2cπtαl)71Γ(1+α)(2cπtαl)-1Γ(1+3α)(2cπtαl)3+1Γ(1+5α)(2cπtαl)5-1Γ(1+7α)(2cπtαl)7+]. In closed form this gives: (35)u(x,y,t)=sin[π(x+yl)]·cos(2cπtαl)-cos[π(x+yl)],sin(2cπtαl)=sin[π(x+y-2ctαl)] which is the exact solution for the given problem. When α=1, the above n-term approximate solution for (30) becomes (36)u(x,y,t)=i=0n-1ui=sin[π(x+yl)]×[1-12!(2cπtl)2+14!(2cπtl)4{(2cπtl)6}-16!(2cπtl)6+]-cos[π(x+yl)]×[(2cπtl)-13!(2cπtl)3{(2cπtl)7}+15!(2cπtl)5-17!(2cπtl)7+]. In closed form, this gives (37)u(x,y,t)=sin[π(x+yl)]·cos(2cπtl)-cos[π(x+yl)],sin(2cπtl)=sin[π(x+y-2ctl)], which is the same result obtained by variational iteration method in .

Example 7.

Consider the following fractional order heat equation in 2-dimensional space : (38)D*tβu(x,y,t)=α(uxx+uyy),u(x,y,0)=c[sin(πxl)+sin(πyl)],0<β1.

The exact solution for this problem when β=1 is (39)u(x,y,t)=ce-απ2t/l2[sin(πxl)+sin(πyl)].

The initial value problem (38) is equivalent to the integral equation (40)u(x,y,t)=c[sin(πxl)+sin(πyl)]+Itβ[α(uxx+uyy)].

Let N(u)=Itβ[α(uxx+uyy)]. In view of recurrence relation (17), we have the following first approximations: (41)u0(x,y,t)=c[sin(πxl)+sin(πyl)],u1(x,y,t)=-cΓ(1+β)(απ2tβl2)×[sin(πxl)+sin(πyl)],u2(x,y,t)=cΓ(1+2β)(απ2tβl2)2×[sin(πxl)+sin(πyl)],u3(x,y,t)=-cΓ(1+3β)(απ2tβl2)3×[sin(πxl)+sin(πyl)], and so on. The n-term approximate solution for (38) is (42)u(x,y,t)=i=0n-1ui=c[sin(πxl)+sin(πyl)]×[1-1Γ(1+β)(απ2tβl2){(απ2tβl2)3}+1Γ(1+2β)(απ2tβl2)2-1Γ(1+3β)(απ2tβl2)3+]. When β=1, The n-term approximate solution for (38) becomes (43)u(x,y,t)=i=0n-1ui=c[sin(πxl)+sin(πyl)]×[13!(απ2tl2)31-11!(απ2tl2)+12!(απ2tl2)2-13!(απ2tl2)3+]. In closed form, this gives (44)u(x,y,t)=ce-απ2t/l2[sin(πxl)+sin(πyl)] which is the exact solution for the given problem.

The obtained results in this example are the same as these obtained in  by the homotopy perturbation method, in case β=1, but with the simplicity of the given method.

Example 8.

In this last example, we consider the following fractional order nonlinear wave equation : (45)D*xαu(x,t)-uutt=x2-αΓ(3-α)-(x2+t2)2,u(0,t)=t22,ux(0,t)=0,1<α2.

The exact solution for this problem when α=2 is u(x,t)=(1/2)(x2+t2) where 0x,  t1.

The initial value problem (45) is equivalent to the integral equation (46)u(x,t)=12(x2+t2)-x2+αΓ(3+α)-t2xα2Γ(1+α)+Ixα[uutt].

Let N(u)=Ixα[uutt]. In view of recurrence relation (8), we have (47)u0(x,t)=12(x2+t2)-x2+αΓ(3+α)-t2xα2Γ(1+α),u1(x,t)=x2+αΓ(3+α)-x2+2αΓ(3+2α)-Γ(3+α)x2+2α2Γ(1+α)Γ(3+2α)+Γ(3+2α)x2+3αΓ(1+α)Γ(3+α)Γ(3+3α)+t2(xα2Γ(1+α)-x2αΓ(1+2α){Γ(1+2α)x3α2Γ(1+α)2Γ(1+3α)}+Γ(1+2α)x3α2Γ(1+α)2Γ(1+3α)), and so on. The 3-term approximate solution and the corresponding exact solution for (45) are plotted in Figure 4(a), in case t=1/2, for  α=1.8,1.9,2., in Figure 4(b), in case t=1, for α=1.8,1.9,2., and in Figure 4(c), in case α=2. It is remarkable to note that in the first two figures all the solutions are almost equal.

(a) Plots of the approximate solution for different values of α and the exact solution, in case t=1/2; for (45). (b) Plots of the approximate solution for different values of α and the exact solution, in case t=1; for (45). (c) Plots of the approximate solution, in case α=2 for (45).

Comparing these results with those obtained by the modification homotopy perturbation method in , we can confirm the accuracy and simplicity of the given method.

5. Conclusion

In this paper, the new iterative method with suitable algorithm is successfully used to solve linear and nonlinear ordinary and partial differential equations with fractional order. It is clear that the computations are easy and the solutions agree well with the corresponding exact solutions and more accurate than the solutions obtained by other methods. Moreover, the accuracy is high with little computed terms of the solution which confirm that this method with the given algorithm is a powerful method for handling fractional differential equations.

Miller K. Ross B. An Introduction to the Fractional Calculus and Fractional Diffrential Equations 1993 New York, NY, USA John Wiley & Sons Oldham K. B. Spanier J. The Fractional Calculus 1974 London, UK Academic Press MR0361633 ZBL0292.26011 Podlubny I. Fractional Differential Equations 1999 198 San Diego, Calif, USA Academic Press MR1658022 ZBL1056.93542 Amairi M. Aoun M. Najar S. Abdelkrim M. N. A constant enclosure method for validating existence and uniqueness of the solution of an initial value problem for a fractional differential equation Applied Mathematics and Computation 2010 217 5 2162 2168 10.1016/j.amc.2010.07.015 MR2727962 ZBL1250.34006 Deng J. Ma L. Existence and uniqueness of solutions of initial value problems for nonlinear fractional differential equations Applied Mathematics Letters 2010 23 6 676 680 10.1016/j.aml.2010.02.007 MR2609797 ZBL1201.34008 Girejko E. Mozyrska D. Wyrwas M. A sufficient condition of viability for fractional differential equations with the Caputo derivative Journal of Mathematical Analysis and Applications 2011 381 1 146 154 10.1016/j.jmaa.2011.04.004 MR2796198 ZBL1222.34007 Ray S. S. Bera R. K. Solution of an extraordinary differential equation by Adomian decomposition method Journal of Applied Mathematics 2004 4 331 338 10.1155/S1110757X04311010 MR2100259 ZBL1080.65069 Dehghan M. Manafian J. Saadatmandi A. Solving nonlinear fractional partial differential equations using the homotopy analysis method Numerical Methods for Partial Differential Equations 2010 26 2 448 479 10.1002/num.20460 MR2605472 ZBL1185.65187 Hashim I. Abdulaziz O. Momani S. Homotopy analysis method for fractional IVPs Communications in Nonlinear Science and Numerical Simulation 2009 14 3 674 684 10.1016/j.cnsns.2007.09.014 MR2449879 ZBL1221.65277 Odibat Z. Momani S. Xu H. A reliable algorithm of homotopy analysis method for solving nonlinear fractional differential equations Applied Mathematical Modelling 2010 34 3 593 600 10.1016/j.apm.2009.06.025 MR2563341 ZBL1185.65139 Hemeda A. A. Homotopy perturbation method for solving partial differential equations of fractional order International Journal of Mathematical Analysis 2012 6 49–52 2431 2448 MR2966013 Hemeda A. A. Homotopy perturbation method for solving systems of nonlinear coupled equations Applied Mathematical Sciences 2012 6 93–96 4787 4800 MR2950641 Esmaeili S. Shamsi M. Luchko Y. Numerical solution of fractional differential equations with a collocation method based on Müntz polynomials Computers & Mathematics with Applications 2011 62 3 918 929 10.1016/j.camwa.2011.04.023 MR2824680 ZBL1228.65132 Hemeda A. A. Variational iteration method for solving wave equation Computers & Mathematics with Applications 2008 56 8 1948 1953 10.1016/j.camwa.2008.04.010 MR2466697 ZBL1165.65396 Hemeda A. A. Variational iteration method for solving non-linear partial differential equations Chaos, Solitons and Fractals 2009 39 3 1297 1303 10.1016/j.chaos.2007.06.025 MR2512934 ZBL1197.35227 Hemeda A. A. Variational iteration method for solving nonlinear coupled equations in 2-dimensional space in fluid mechanics International Journal of Contemporary Mathematical Sciences 2012 7 37–40 1839 1852 MR2958998 ZBL1254.76110 Sakar M. G. Erdogan F. Yıldırım A. Variational iteration method for the time-fractional Fornberg-Whitham equation Computers & Mathematics with Applications 2012 63 9 1382 1388 10.1016/j.camwa.2012.01.031 MR2912063 ZBL1247.65138 Doha E. H. Bhrawy A. H. Ezz-Eldien S. S. Efficient Chebyshev spectral methods for solving multi-term fractional orders differential equations Applied Mathematical Modelling 2011 35 12 5662 5672 10.1016/j.apm.2011.05.011 MR2820942 ZBL1228.65126 Doha E. H. Bhrawy A. H. Ezz-Eldien S. S. A Chebyshev spectral method based on operational matrix for initial and boundary value problems of fractional order Computers & Mathematics with Applications 2011 62 5 2364 2373 10.1016/j.camwa.2011.07.024 MR2831698 ZBL1231.65126 Daftardar-Gejji V. Jafari H. An iterative method for solving nonlinear functional equations Journal of Mathematical Analysis and Applications 2006 316 2 753 763 10.1016/j.jmaa.2005.05.009 MR2207344 ZBL1087.65055 Hemeda A. A. New iterative method: application to nth-order integro-differential equations International Mathematical Forum 2012 7 47 2317 2332 Hemeda A. A. Formulation and solution of nth-order derivative fuzzy integrodifferential equation using new iterative method with a reliable algorithm Journal of Applied Mathematics 2012 2012 17 10.1155/2012/325473 325473 MR2979467 ZBL1251.34015 Saadatmandi A. Dehghan M. A new operational matrix for solving fractional-order differential equations Computers & Mathematics with Applications 2010 59 3 1326 1336 10.1016/j.camwa.2009.07.006 MR2579495 ZBL1189.65151 Doha E. H. Bhrawy A. H. Ezz-Eldien S. S. A new Jacobi operational matrix: an application for solving fractional differential equations Applied Mathematical Modelling 2012 36 10 4931 4943 10.1016/j.apm.2011.12.031 Ghazanfari B. Ghazanfari A. G. Fuladvand M. Modification of the homotopy perturbation method for numerical solution of nonlinear wave and system of nonlinear wave equations The Journal of Mathematics and Computer Science 2011 3 2 212 224 Caputo M. Linear methods of dissipation whose Q is almost frequency independent, part II Journal of the Royal Society of Medicine 1967 13 529 539 Cherruault Y. Convergence of Adomian's method Kybernetes 1989 18 2 31 38 10.1108/eb005812 MR1009979 ZBL0697.65051 Jerri A. J. Introduction to Integral Equations with Applications 1999 2nd New York, NY, USA Wiley-Interscience MR1800272 Bhalekar S. Daftardar-Gejji V. Convergence of the new iterative method International Journal of Differential Equations 2011 2011 10 989065 MR2854946 ZBL1239.34014