# Least Squares Differential Quadrature Method for the Generalized Bagley–Torvik Fractional Differential Equation

Authors: Constantin Bota; Bogdan Căruntu; Mădălina Sofia Paşca; Dumitru Ţucu; Marioara Lăpădat
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the least squares differential quadrature method for computing approximate analytical solutions for the generalized Bagley–Torvik fractional differential equation is presented. This new method is introduced as a straightforward and accurate method, fact proved by the examples included, containing a comparison with previous results obtained by using other methods.


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## Body

## 1. Introduction

Even though the fractional calculus is a relatively old concept, dating back from the time of the mathematicians L’Hospital and Leibniz in the 17th century, it has seen a significant development in the last decades only. The fractional differential equations have numerous applications in biology, physics, chemistry, and engineering, which is why many scientists are concerned about finding more effective ways to solve them. Unfortunately, finding analytical solutions for fractional differential equations is often difficult or even impossible, which is why in recent years various numerical and approximate methods have been developed. One of the fractional differential equations with wide applicability in engineering is the Bagley–Torvik equation. This equation was introduced by Bagley and Torvik in [1] in 1984 to “model a motion of a rigid plate immersed in a Newtonian fluid” and in [2] for modeling the damping properties of the polymers. In engineering (construction, biotechnology, and chemistry), condensing polymers are widely used, and in their study, the Bagley–Torvik equation is the one employed [3, 4].Bagley–Torvik-type fractional differential equations have been studied both numerically and analytically in numerous articles. Among the methods used to solve this equation, we mention the following: numerical methods [5], Adomian decomposition method [6], discrete spline methods [7], Haar wavelet method [8, 9], homotopy perturbation method [10], sinc collocation method [11, 12], cubic spline method [13], quadratic spline solution [14], B-spline collocation method [15, 16], Chebyshev collocation method [17], hybrid functions approximation [18], harmonic wavelets [19], predictor-corrector method of Adams type [20], spectral methods [21], fractional natural decomposition method [22], and finite element method [23].The motivation of this paper is to introduce a new method for obtaining analytical approximate solutions for fractional differential equations. This method is obtained as a combination of the differential quadrature method (DQM) ([24]) and the least squares method (LSM), and we named it the least squares differential quadrature method (LSDQM). We applied this new method to find solutions of the following generalized Bagley–Torvik fractional differential equation [25]:(1)A⋅y′′x+Bx⋅Dxαyx=fx,yx,Dxβyx,y′x,where yx:a,b⟶ℝ, α∈1,2,β∈0,1, together with the conditions:(2)ya=μ0,ν1⋅y′a+ν2⋅yb=μ1,where μ0, μ1, ν0, and ν1 are constants (for ν1=0, we obtain boundary conditions, while for ν2=0, we obtain initial conditions). A∈ℝ, Bx and fx are given such that problems (1) and (2) satisfy the existence and uniqueness conditions for a continuous solution [25].Dxγ denotes Caputo’s fractional derivative:(3)Dxγyx=1Γn−γ⋅∫0xx−ζ−γ−n+1⋅ynζdζ,n−1<γ≤n,n∈ℕ∗.In the next section, we will present the least squares differential quadrature method (LSDQM), and in the third section, some numerical examples are presented followed by the conclusions.


## 2. The Least Squares Differential Quadrature Method

First, we will consider a numerical mesh of the intervalI=a,b by means of a partition ΔM consisting of M+1 points: a=x0<x1<x2<⋯<xM−1<xM=bp=0.007. In order to find an approximate solution of the generalized Bagley–Torvik problems (1) and (2), we will compute the values of a certain functional (the functional (12) introduced in the following) at the point xi.To equation (1), we attach the following operator:(4)Dαyx=A⋅y″x+Bx⋅Dxαyx−fx,yx,Dxβyx,y′x.We denote byy˜x an approximate solution of equation (1). By replacing in Dα the exact solution yx with this approximate solution, we obtain the following reminder:(5)ℛx,y˜x=Dαy˜x.Definition 1.
We call anϵ-approximate solution of problems (1) and (2) related to the partition ΔM an approximate polynomial solution, which satisfies the following relations:(6)ℛxi,y˜xi|<ε,i=0,M¯,(7)y˜a=μ0,ν1⋅y˜′a+ν2⋅y˜b=μ1.Definition 2.
We consider the sequence of polynomials:(8)PNx=∑k=0Nckxk,ck∈ℝ,k=0,N¯.
We call the sequence of polynomialsPNx convergent to the solution of problems (1) and (2) if(9)limN⟶∞DαPNx=0.
Taking into account the above definitions, we will computeϵ-approximate polynomial solutions for a Bagley–Torvik problem of the types (1) and (2) by taking the steps described in the following algorithm:(i)
Step 1. we will compute an approximate solution of the type(10)TNx=∑k=0N˜ckxk,where the sequence of polynomials must also satisfy the boundary conditions:(11)TNa=μ0,ν1⋅TN′a+ν2⋅TNb=μ1,and the constantsc˜k are calculated in the following steps.(ii)
Step 2. From the boundary conditions, we obtain ˜c0 and ˜c1 as functions of ˜c2,˜c3,…,c˜N. Thus, the expression of TNx from now on will be a function of x and c˜2,c˜3,…,c˜N only.(iii)
Step 3. We attach to problems (1) and (2) the functional ℱc˜2,c˜3,…,c˜N:(12)ℱ=∑i=0Mℛ2xi,TNxi==∑i=0MA⋅TN″xi+Bx⋅DxαTNxi−fx,TNxi,Dxβyx,TN′xi2.(iv)
Step 4. By minimizing the functional (12), we obtain the values of the coefficients c˜20,c˜30,…,c˜N0 which give the minimum of (12).(v)
Step 5. After computing c˜20,c˜30,…,c˜N0, we return to the boundary conditions and obtain the final value of c0˜ and c1˜.(vi)
Step 6. Finally, we replace the values c˜00,c˜10,c˜20,c˜30,…,c˜N0, calculate in the expression of TNx, and denote by TN0x=∑k=0Nc˜k0xk the analytical approximate polynomial solution by LSDQM of problems (1) and (2).
The following convergence theorem is satisfied:Theorem 1.
The sequence of polynomialsTNx satisfies the following relation:(13)limN⟶∞ℛ2xi,TNxi=0,i=0,M¯.Proof.
Letyx be an exact solution of problems (1) and (2), which means from hypothesis that there exists a sequence of polynomials PNx=∑k=0Nckxk,\\ck∈ℝ,k=0,N¯ converging to yx (according to the Weierstrass theorem on Polynomial Approximation, see [24]), i.e., limN⟶∞PNx=yx,∀x∈I. Taking into account Definition 2, this means that PNx satisfies(14)limN⟶∞DαPNx=0.
Denoting byc20,c30,…,cN0 the values of the coefficients ck which give the minimum of the functional (12), it follows that ℱc20,c30,…,cN0≤ℱc2,c3,…,cN. If TN0x=∑k=0Nc˜k0xk is the approximate solution by LSDQM (with c˜00 and c˜10 computed by using the initial conditions), we obtain(15)∑i=0Mℛ2xi,TN0xi≤∑i=0Mℛ2xi,PNxi.Hence, limN⟶∞∑i=0Mℛ2xi,TN0xi≤≤limN⟶∞∑i=0Mℛ2xi,PNxi.
We conclude thatlimN⟶∞ℛ2xi,TN0xi=0.


### 2.1. Error Estimation

We can rewrite equation (1) as(16)A⋅y″x+Bx⋅Dxαyx−f1yx,Dxβyx,yx=f2x.We denote withℒ the following operator:(17)ℒyx=A⋅y″x+Bx⋅Dxαyx−f1yx,Dxβyx,y′x.Thus, equation (1) becomes(18)ℒyx=f2x.Using (for simplicity) the notation:ℛx=ℛx,y˜x=Dαy˜x, in (4), we obtain(19)ℛx=ℒy˜x−f2x,with y˜ as an approximate solution for problems (1) and (2), which means as y˜ satisfies(20)ℒy˜x=A⋅y″x+Bx⋅Dxαyx−f1yx,Dxβyx,y′x=f2x+ℛx,together with the conditions:(21)y˜a=μ0,ν1⋅y˜′a+ν2⋅y˜b=μ1.We define the error function in the following way:e˜x=yx−y˜x, where yx is the exact solution for problems (1) and (2) and we obtain (using (18) and (19)) the differential equation for the error function:(22)ℒe˜x=ℒyx−ℒy˜x=−ℛx,with the conditions:(23)e˜a=0,ν1⋅e˜′a+ν2⋅e˜b=0.The problem for the error function becomes(24)ℒe˜x=−ℛx.Solving (24) together with (23) in the same manner as described above for problems (1) and (2), we obtain the approximation e˜x. We will be able to determine the absolute maximum error:(25)E=maxe˜x,0≤x≤1.In this manner, we can estimate the error without knowing the exact solution of the initial problems (1) and (2).Remark 1.
We remark that the above theorem proves the convergence of LSDQM since as the degree of the polynomial increases the remainder corresponding to the approximation tends to zero. This fact is illustrated in the second example.


## 2.1. Error Estimation

We can rewrite equation (1) as(16)A⋅y″x+Bx⋅Dxαyx−f1yx,Dxβyx,yx=f2x.We denote withℒ the following operator:(17)ℒyx=A⋅y″x+Bx⋅Dxαyx−f1yx,Dxβyx,y′x.Thus, equation (1) becomes(18)ℒyx=f2x.Using (for simplicity) the notation:ℛx=ℛx,y˜x=Dαy˜x, in (4), we obtain(19)ℛx=ℒy˜x−f2x,with y˜ as an approximate solution for problems (1) and (2), which means as y˜ satisfies(20)ℒy˜x=A⋅y″x+Bx⋅Dxαyx−f1yx,Dxβyx,y′x=f2x+ℛx,together with the conditions:(21)y˜a=μ0,ν1⋅y˜′a+ν2⋅y˜b=μ1.We define the error function in the following way:e˜x=yx−y˜x, where yx is the exact solution for problems (1) and (2) and we obtain (using (18) and (19)) the differential equation for the error function:(22)ℒe˜x=ℒyx−ℒy˜x=−ℛx,with the conditions:(23)e˜a=0,ν1⋅e˜′a+ν2⋅e˜b=0.The problem for the error function becomes(24)ℒe˜x=−ℛx.Solving (24) together with (23) in the same manner as described above for problems (1) and (2), we obtain the approximation e˜x. We will be able to determine the absolute maximum error:(25)E=maxe˜x,0≤x≤1.In this manner, we can estimate the error without knowing the exact solution of the initial problems (1) and (2).Remark 1.
We remark that the above theorem proves the convergence of LSDQM since as the degree of the polynomial increases the remainder corresponding to the approximation tends to zero. This fact is illustrated in the second example.


## 3. Numerical Examples

In this section, we will present some examples of problems including generalized Bagley–Torvik fractional differential equations together with boundary or initial conditions and problems solved using the least squares differential quadrature method (LSDQM).


### 3.1. Example 1

We consider the generalized Bagley–Torvik equation [26, 27]:(26)y″x+x1/2⋅Dαyx+Byx=−x1/3⋅yx−x1/4⋅Dβyx−x1/5yx+f2x,together with the initial conditions: y0=2,y′0=0, where f2x=1−x1/2/Γ3−α⋅x2−α−x1/3⋅x−x1/4/Γ3−β⋅x2−β+x1/5⋅2−1/2⋅x2, α=1.234, and β=0.333. The exact solution of this problem is yx=2−1/2⋅x2.Approximate solutions for this problem with absolute errors larger than10−4 were proposed by El-Mesiry et al. in [26] and with absolute errors larger than 10−5 by Li and Zhao in [27].We applied LSDQM to find an approximate analytical solution on the0,1 interval for (26).In Step 1, we choose an approximate solutiony˜x of the following type:(27)y˜x=c˜0+c˜1x+c˜2x2.In Step 2, by using the initial conditions, we findc˜0=2 and c˜1=0, and the approximate solution becomes(28)y˜x=2+c˜2x2.In Step 3, the corresponding reminder (5) is(29)ℛy˜=c˜22x4/3+x11/5+1000x633/500383Γ383/500+2000000x1917/10001111889Γ667/1000+2+x4/3+x11/52+x633/500Γ883/500+x1917/1000Γ2667/1000+1.We consider the simplest possible partitionΔ1: 0=x0<x1=1, and we attach to the equation the functional ℱc˜2=∑i=01ℛ2xi,T1xi (too long to be included here).In Step 4, in order to find the minimum of this functional, we can compute the stationary points by equating to zero its first derivative (the computations are performed using the software Wolfram Mathematica). We obtainc˜2=1/2, and it is easy to show (by means of the second derivative) that this stationary point is indeed the minimum.In the case of this problem, Step 5 is not needed since we already have the final values ofc˜0 and c˜1.Finally, in Step 6, we replacec˜0=2, c˜1=0, and c˜2=1/2 in the initial expression of y˜x, thus obtaining the exact solution of the equation: y˜x=2−1/2⋅x2.We remark that, in the case of this example, we could choose any partitionΔM since the particular form of the remainder (5) assures the fact that the stationary point is always c˜2=1/2 no matter how large M is.


### 3.2. Example 2

We consider the fractional Bagley–Torvik boundary value problem [21]:(30)y″x+Dαyx+yx=f2x,(31)f2x=−4c3x13/2F21;5/4,7/4;−1/4c2x23π−c2sincx+sincx,(32)y0=0,y1=sinc,where a,b=0,1, α=1.5, and F is the generalized hypergeometric function.The exact solution of this problem isyx=sinc⋅x.We remark the fact that if the exact solutions of the problem is a polynomial function (as in the case of the first example), then, naturally, in applying the first step of our method, we have chosen an approximate polynomial solution (10) of the same degree as the exact solution. Since, in this example, the exact solution is not a polynomial, we computed approximate polynomial solutions of several degrees, and for each degree, we calculated the maximal value of the absolute error on the whole interval 0,1 for c=1. The results presented in Table 1 together with a comparison with previous results from [21] clearly illustrate the convergence of the method.Table 1
Maximal absolute errors for approximate polynomial solutions of various degrees obtained using LSDQM for example 2.
DegreeMaximal error [21]Maximal error LSDQM43.4⋅10−46.3979⋅10−55—1.3366⋅10−66—8.2879⋅10−87—1.3333⋅10−984.3⋅10−77.3808⋅10−119—1.0573⋅10−1210—6.5171⋅10−14The approximate polynomial solutions computed by LSDQM using an equidistant partitionΔ100 are(33)y˜4x=0.0196467x4−0.181975x3+0.00382023x2+0.999979x,(34)y˜5x=0.00720937x5+0.00159936x4−0.167467x3+0.000129252x2+1.x,(35)y˜6x=−0.000658967x6+0.00918884x5−0.000526231x4−0.166518x3−0.0000146541x2+1.x,(36)y˜7x=−0.000172399x7−0.0000548963x6+0.00838255x5−0.0000217806x4−0.166662x3−3.209289⋅10−7x2+1.x,(37)y˜8x=0.0000117989x8−0.000219638x7+0.0000202955x6+0.00832241x5+3.233537⋅10−6x4−0.166667x3+2.3498896⋅10−8x2+1.x,(38)y˜9x=2.40043968⋅10−6x9+9.888419⋅10−7x8−0.000199677x7+9.028126⋅10−7x6+0.00833296x5+8.6069828⋅10−8x4−0.166667x3(39)+3.73299399⋅10−10x2+1.x,(40)y˜10x=−1.3736474⋅10−7x10+3.0878011⋅10−6x9−4.5494814⋅10−7x8−0.000198028x7−2.0494998⋅10−7x6+0.0083334x5−1.31793399⋅10−8x4−0.166667x3−4.5065671⋅10−11x2+1.x.


### 3.3. Example 3

Choosing in (1) A=0, Bx=0, fx,yx,Dxβyx,y′x=Dxβyx+y2x−1, a,b=0,1, μ0=μ1=ν0=ν1=0, and α∈0,1, we obtain the problem [5, 28]:(41)Dxβyx+y2x−1=0,(42)y0=0.This equation is actually a Riccati-type equation, and forα=1, the exact solution of the problem is yx=e2x−1/e2x+1.Following the LSDQM steps presented in the previous section, we computed approximate polynomial solutionsy˜x of the 9th degree for problems (41) and (42) choosing in turn β=1, β=0.9, β=0.8, and β=0.7. The solutions are computed on the 0,1 interval using again an equidistant partition Δ100. The results are presented in Figure 1.Figure 1
Approximate solutions for problems (41) and (42) corresponding to several values of β.For example, the approximate polynomial solutions corresponding toβ=1 is(43)y˜x=−0.0106498⋅x9+0.0567197⋅x8−0.103159⋅x7+0.0253074⋅x6++0.125356⋅x5+0.00150329⋅x4−0.333491⋅x3+7.88553637⋅10−6⋅x2+1⋅x.The maximal absolute error corresponding to this approximation is4.11⋅10−9.We remark that the results presented in Figure1 are in good agreement with previously obtained results [5].


### 3.4. Example 4

While LSDQM, as most methods of this type, works best on small intervals, it can also be successfully employed in the case of large intervals. In order to illustrate this feature, we consider the Bagley–Torvik problem [6]:(44)y″x+Dxαyx+yx=0,(45)y0=1,y′0=0,where α∈1,2.The exact solution of this problem is not known.An approximate solution obtained by using the Adomian decomposition method was proposed in [6] by V. Daftardar-Gejji and H. Jafari.Forα=1.5, using an equidistant partition Δ10 on the interval I=0,20, we obtain the approximate analytical solution, presented in Figure 2:(46)y˜=−4.24381⋅10−9⋅x9+3.757992⋅10−7⋅x8−0.0000130166⋅x7+0.00021244,(47)x6−0.00133775⋅x5−0.00503943⋅x4+0.108157⋅x3−0.404114⋅x2+1.Figure 2
Approximate solution for the problems (44) and (45) corresponding to α=1.5.We remark that our result is in very good agreement with the one in [6].


## 3.1. Example 1

We consider the generalized Bagley–Torvik equation [26, 27]:(26)y″x+x1/2⋅Dαyx+Byx=−x1/3⋅yx−x1/4⋅Dβyx−x1/5yx+f2x,together with the initial conditions: y0=2,y′0=0, where f2x=1−x1/2/Γ3−α⋅x2−α−x1/3⋅x−x1/4/Γ3−β⋅x2−β+x1/5⋅2−1/2⋅x2, α=1.234, and β=0.333. The exact solution of this problem is yx=2−1/2⋅x2.Approximate solutions for this problem with absolute errors larger than10−4 were proposed by El-Mesiry et al. in [26] and with absolute errors larger than 10−5 by Li and Zhao in [27].We applied LSDQM to find an approximate analytical solution on the0,1 interval for (26).In Step 1, we choose an approximate solutiony˜x of the following type:(27)y˜x=c˜0+c˜1x+c˜2x2.In Step 2, by using the initial conditions, we findc˜0=2 and c˜1=0, and the approximate solution becomes(28)y˜x=2+c˜2x2.In Step 3, the corresponding reminder (5) is(29)ℛy˜=c˜22x4/3+x11/5+1000x633/500383Γ383/500+2000000x1917/10001111889Γ667/1000+2+x4/3+x11/52+x633/500Γ883/500+x1917/1000Γ2667/1000+1.We consider the simplest possible partitionΔ1: 0=x0<x1=1, and we attach to the equation the functional ℱc˜2=∑i=01ℛ2xi,T1xi (too long to be included here).In Step 4, in order to find the minimum of this functional, we can compute the stationary points by equating to zero its first derivative (the computations are performed using the software Wolfram Mathematica). We obtainc˜2=1/2, and it is easy to show (by means of the second derivative) that this stationary point is indeed the minimum.In the case of this problem, Step 5 is not needed since we already have the final values ofc˜0 and c˜1.Finally, in Step 6, we replacec˜0=2, c˜1=0, and c˜2=1/2 in the initial expression of y˜x, thus obtaining the exact solution of the equation: y˜x=2−1/2⋅x2.We remark that, in the case of this example, we could choose any partitionΔM since the particular form of the remainder (5) assures the fact that the stationary point is always c˜2=1/2 no matter how large M is.


## 3.2. Example 2

We consider the fractional Bagley–Torvik boundary value problem [21]:(30)y″x+Dαyx+yx=f2x,(31)f2x=−4c3x13/2F21;5/4,7/4;−1/4c2x23π−c2sincx+sincx,(32)y0=0,y1=sinc,where a,b=0,1, α=1.5, and F is the generalized hypergeometric function.The exact solution of this problem isyx=sinc⋅x.We remark the fact that if the exact solutions of the problem is a polynomial function (as in the case of the first example), then, naturally, in applying the first step of our method, we have chosen an approximate polynomial solution (10) of the same degree as the exact solution. Since, in this example, the exact solution is not a polynomial, we computed approximate polynomial solutions of several degrees, and for each degree, we calculated the maximal value of the absolute error on the whole interval 0,1 for c=1. The results presented in Table 1 together with a comparison with previous results from [21] clearly illustrate the convergence of the method.Table 1
Maximal absolute errors for approximate polynomial solutions of various degrees obtained using LSDQM for example 2.
DegreeMaximal error [21]Maximal error LSDQM43.4⋅10−46.3979⋅10−55—1.3366⋅10−66—8.2879⋅10−87—1.3333⋅10−984.3⋅10−77.3808⋅10−119—1.0573⋅10−1210—6.5171⋅10−14The approximate polynomial solutions computed by LSDQM using an equidistant partitionΔ100 are(33)y˜4x=0.0196467x4−0.181975x3+0.00382023x2+0.999979x,(34)y˜5x=0.00720937x5+0.00159936x4−0.167467x3+0.000129252x2+1.x,(35)y˜6x=−0.000658967x6+0.00918884x5−0.000526231x4−0.166518x3−0.0000146541x2+1.x,(36)y˜7x=−0.000172399x7−0.0000548963x6+0.00838255x5−0.0000217806x4−0.166662x3−3.209289⋅10−7x2+1.x,(37)y˜8x=0.0000117989x8−0.000219638x7+0.0000202955x6+0.00832241x5+3.233537⋅10−6x4−0.166667x3+2.3498896⋅10−8x2+1.x,(38)y˜9x=2.40043968⋅10−6x9+9.888419⋅10−7x8−0.000199677x7+9.028126⋅10−7x6+0.00833296x5+8.6069828⋅10−8x4−0.166667x3(39)+3.73299399⋅10−10x2+1.x,(40)y˜10x=−1.3736474⋅10−7x10+3.0878011⋅10−6x9−4.5494814⋅10−7x8−0.000198028x7−2.0494998⋅10−7x6+0.0083334x5−1.31793399⋅10−8x4−0.166667x3−4.5065671⋅10−11x2+1.x.


## 3.3. Example 3

Choosing in (1) A=0, Bx=0, fx,yx,Dxβyx,y′x=Dxβyx+y2x−1, a,b=0,1, μ0=μ1=ν0=ν1=0, and α∈0,1, we obtain the problem [5, 28]:(41)Dxβyx+y2x−1=0,(42)y0=0.This equation is actually a Riccati-type equation, and forα=1, the exact solution of the problem is yx=e2x−1/e2x+1.Following the LSDQM steps presented in the previous section, we computed approximate polynomial solutionsy˜x of the 9th degree for problems (41) and (42) choosing in turn β=1, β=0.9, β=0.8, and β=0.7. The solutions are computed on the 0,1 interval using again an equidistant partition Δ100. The results are presented in Figure 1.Figure 1
Approximate solutions for problems (41) and (42) corresponding to several values of β.For example, the approximate polynomial solutions corresponding toβ=1 is(43)y˜x=−0.0106498⋅x9+0.0567197⋅x8−0.103159⋅x7+0.0253074⋅x6++0.125356⋅x5+0.00150329⋅x4−0.333491⋅x3+7.88553637⋅10−6⋅x2+1⋅x.The maximal absolute error corresponding to this approximation is4.11⋅10−9.We remark that the results presented in Figure1 are in good agreement with previously obtained results [5].


## 3.4. Example 4

While LSDQM, as most methods of this type, works best on small intervals, it can also be successfully employed in the case of large intervals. In order to illustrate this feature, we consider the Bagley–Torvik problem [6]:(44)y″x+Dxαyx+yx=0,(45)y0=1,y′0=0,where α∈1,2.The exact solution of this problem is not known.An approximate solution obtained by using the Adomian decomposition method was proposed in [6] by V. Daftardar-Gejji and H. Jafari.Forα=1.5, using an equidistant partition Δ10 on the interval I=0,20, we obtain the approximate analytical solution, presented in Figure 2:(46)y˜=−4.24381⋅10−9⋅x9+3.757992⋅10−7⋅x8−0.0000130166⋅x7+0.00021244,(47)x6−0.00133775⋅x5−0.00503943⋅x4+0.108157⋅x3−0.404114⋅x2+1.Figure 2
Approximate solution for the problems (44) and (45) corresponding to α=1.5.We remark that our result is in very good agreement with the one in [6].


## 4. Conclusions

In this paper, the application of the least squares differential quadrature method (LSDQM) to the Bagley–Torvik fractional differential equation is presented.Due to the fact that the method is relative straightforward, the approximations may be obtained in a quick and simple manner. The numerical examples included clearly illustrate the accuracy of the method by means of comparisons with solutions previously computed by other methods. If the exact solution of the problem is itself a polynomial, then usually LSDQM is able to find the exact solution, as in the case of the first example. If not, in most of the cases, the solutions provided by LSDQM are not only more precise, but being of the simplest form (i.e., polynomial solutions), they are much easier to use in subsequent computation than many of the solutions provided by other methods.



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*Source: 4806387-2020-07-16.xml*