# Numerical Solution for Third-Order Two-Point Boundary Value Problems with the Barycentric Rational Interpolation Collocation Method

Authors: Qian Ge; Xiaoping Zhang
Journal: Journal of Mathematics
 (2021)

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## Abstract

The numerical solution for a kind of third-order boundary value problems is discussed. With the barycentric rational interpolation collocation method, the matrix form of the third-order two-point boundary value problem is obtained, and the convergence and error analysis are obtained. In addition, some numerical examples are reported to confirm the theoretical analysis.


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## Body

## 1. Introduction

Differential equations can give full play to their mathematical advantages in various disciplines. Combining the theory of differential equations with practical problems can build models of practical problems. Many engineering and physical problems can be transformed into the initial boundary value problems of differential equations. In these problems, only a few simple cases can be solved analytically, and most engineering problems need to be solved by numerical methods. Compared with polynomial interpolation, rational function interpolation has higher interpolation accuracy and can effectively overcome the instability of interpolation [1–4]. Barycentric rational interpolation not only has high interpolation accuracy on special distributed nodes but also has high interpolation accuracy for equidistant nodes [5–7]. This method has been used to solve certain problems such as Volterra integral equations [2, 8, 9], delay Volterra integrodifferential equations [10, 11], plane elastic problems [12], nonlinear problems [13], heat conduction equation [14], and so on [15–17].The third-order differential equation has a wide range of applications and important theoretical values in many scientific fields, such as applied mathematics and physics. Therefore, the third-order boundary value problem has been widely concerned by many scholars [18–20]. In this paper, we consider the numerical solution of the third-order two-point boundary value problem,(1)u‴x+pu″x+qu′x+rux=fx,a<x<b,(2)ua=A,u′a=B,u′b=C,orua=A,u′a=B,u″a=C,by the barycentric rational interpolation collocation method.Barycentric rational interpolation collocation method means using barycentric interpolation polynomials to find the differential matrix of a function at each discrete point; thus, the solution of the differential equation can be obtained by matrix operation. The barycentric rational interpolation has excellent numerical stability and high approximation accuracy, and the barycentric rational interpolation formula has a compact calculation formula of all order derivatives. Therefore, the barycentric rational interpolation collocation method is an effective method for solving boundary value problems of differential equations.


## 2. Formula of the Barycentric Interpolation Collocation Method

Discretize the intervala,b into n uniform parts with h=b−a/n, and suppose u1,u2,…,un is the function value of an unknown function u at discrete nodes x1,x2,…,xn.For any0≤d≤n, Pxi,i=0,1,…,n−d, is the interpolation function at the point xi,xi+1,…,xi+d; then, we have Pixk=fxk,k=i,i+1,…,i+d, and(3)rx=∑i=0n−dλixPix∑i=0n−dλix,where(4)λix=−1ix−xi⋯x−xi+d.By changing the polynomialPix into the Lagrange interpolation form as(5)Pix=∑k=ii+d∏j=i,j≠ki+dx−xjxk−xjfkand combining (4) and (5) together, we get(6)∑i=0n−dλixPix=∑i=0n−d−1i∑k=ii+d1x−xk∏j=i,j≠ki+d1xk−xjfk=∑k=0nwkx−xkfk,where wk=∑i∈Jk−1i∏j=i,j≠ki+d1/xk−xj,Jk=i∈I;k−d≤i≤k.Then, we get(7)rx=∑j=0nwj/x−xjfj∑j=0nwj/x−xj,where its basis function is(8)Ljx=wj/x−xj∑k=0nwk/x−xk.For the equidistant point, the weight function is(9)wj=−1n−jCnj.For the Chebyshev point of the second kind,(10)xj=cosjπn,j=0,1,…,n,the weight function is(11)wj=−1jδj,δj=12,j=0,n,1,otherwise.By formula (8), the m-th order derivative of ux at the nodes x1,x2,…,xn can be expressed as(12)umxj:=uim=dmuxjdxm=∑k=1nLkmxjuk=∑k=1nDijmuk,m=1,2,…,and then (12) can be written in the matrix form as(13)um=Dmu,where um=u1m,u2m,…,unmT and u=u1,u2,…,unT.By using the barycentric interpolation function as(14)unx=∑j=0nLjxuj,equation (1) can be written in the numerical form as(15)∑j=0nujLj″′x+p∑j=0nujLj″x+q∑j=0nujLj′x+r∑j=0nujLjx=fx.By using the notation of the differential matrix, (15) can also be denoted as(16)∑j=0nDij3uj+p∑j=0nDij2uj+q∑j=0nDij1uj+r∑j=0nδijuj=fxi,i=1,2,…,n,or the simple matrix form(17)D3+pD2+qD1+rIu=f.Boundary conditions (2) can be divided into(18)u1=A,u′x1=∑j=0nD1j1uj=B,u′xn=∑j=0nDnj1uj=C,oru1=A,u′x1=∑j=0nD1j1uj=B,u″x1=∑j=0nD1j2uj=C,where Dk=Dijkn+1×n+1,(19)Dij1=wixi−xj,i≠j,−∑k≠iDik1,i=j,Dij2=2Dii1Dij1−Dij1xi−xj,i≠j,−∑k≠iDik2,i=j,Dij3=3Dii2Dij1−Dij2xi−xj,i≠j,−∑k≠iDik3,i=j.u=u0,u1,…,unT,f=fx0,fx1,…,fxnT.


## 3. Convergence and Error Analysis

In this section, we will consider the error problem of equidistant interpolation nodes:(20)xi=a+b−ani,i=0,1,…,n.Letux be the solution of (1); for any 0≤d≤n, suppose Pxi,i=0,1,…,n−d, to be the barycentric interpolation function at the point xi,xi+1,…,xi+d; then, we have Pixk=fxk,k=i,i+1,…,i+d, and(21)rx=∑i=0n−dλixPix∑i=0n−dλix,where(22)λix=−1ix−xi⋯x−xi+d.Then, the error function is defined as(23)ex:=ux−Px=x−xi⋯x−xi+dxi,xi+1,…,xi+d,xf,and(24)ex=∑i=0n−dλixux−Pix∑i=0n−dλix=AxBx=Ohd+1,where Ax:=∑i=0n−d−1ixi,…,xi+d,xfandBx:=∑i=0n−dλix.Taking the numerical form,(25)∑j=0nujLj″′x+p∑j=0nujLj″x+q∑j=0nujLj′x+r∑j=0nujLjx=fx,and combining (24) and (1), we get(26)e″′x+pe″x+qe′x+rex=Rfx,where Rfx=fx−fxk,k=0,1,2,…,n.Lemma 1.
Forex defined in (23), we have(27)ex≤Chd+1,u∈Cd+2a,b,e′x≤Chd,u∈Cd+2a,b,e″x≤Chd−1,u∈Cd+3a,b,d≥1,e″′x≤Chd−2,u∈Cd+4a,b,d≥2.Letux be the solution of (1) and unx be the numerical solution; then, we have(28)un″′xk+pun″xk+qun′xk+runxk=fxk,k=0,1,2,…,…n…,and(29)limn⟶∞unx=ux.The results can be obtained in [1].Based on Lemma1, we can get the following theorem.Theorem 1.
Letfx∈Ca,b, Tux:=u″′x+pu″x+qu′x+rux, and unx:Tunx=fx,un∗x:Tun∗x=f∗x; then, we have(30)unx−un∗x≤Chd−2.Proof.
LetL:=D3+pD2+qD1+rI(31)=D003+pD002+qD001+rD013+pD012+qD011⋯D0n3+pD0n2+qD0n1D103+pD102+qD101D113+pD112+qD111+r⋯D1n3+pD1n2+qD1n1⋯⋯⋯⋯Dn03+pDn02+qDn01Dn13+pDn12+qDn11⋯Dnn3+pDnn2+qDnn1+r.
Add column 2, column 3,…, columnn to column 1, and we have(32)L=rD013+pD012+qD011⋯D0n3+pD0n2+qD0n1rD113+pD112+qD111+r⋯D1n3+pD1n2+qD1n1⋯⋯⋯⋯rDn13+pDn12+qDn11⋯Dnn3+pDnn2+qDnn1+r.
Then, we haveL≠0 with r≠0, unx=∑j=0nLjxfj,andun∗x=∑j=0nLjxfj∗, where Un=fx0,fx1,…,fxnTandUn∗=f∗x0,f∗x1,…,f∗xnT.
By(33)Un−Un∗=L−1LUn−Fn∗,which means(34)unx−un∗x=∑MjxTex,where Mjx is the element of matrix L−1, we have(35)unx−un∗x≤∑MjxTex≤Chd−2.
The proof is completed.


## 4. Numerical Example

As an example, we consider the two-point boundary value problem:(36)y″′+y=fx,−1<x<1,(37)y−1=0,y1=0,y′−1=0.For this problem, we can find a functionfx such that the analysis solution is(38)y=1−x21+xeλx,where λ is a freely selected parameter.Substituting (38) into (36), we get(39)fx=−6−6λ1+3x+3λ21−2x−3x2+λ31+x−x2−x3+1−x21+xeλx.For different values ofd and different number of nodes, we can calculate the corresponding relative error and convergence rate; some of the data are shown in Tables 1 and 2.Table 1
Errors and convergence rate of the equidistant nodes with differentd.
λ=2nd=2d=3d=4d=5ErrorhαErrorhαErrorhαErrorhα107.1976e + 004.6412e + 002.4217e + 001.1522e + 00203.5555e + 001.01751.2336e + 001.91173.6419e −012.73339.9594e −023.5322401.4588e + 001.28532.6378e −012.22544.1431e −023.13596.0309e −034.0456805.5050e −011.40605.1045e −022.36954.1384e −033.32363.1073e −044.27861602.0058e −011.45669.4229e −032.43753.8810e −043.41461.4799e −054.39213207.1879e −021.48061.7004e −032.47033.5301e −053.45866.7836e −074.44736402.5569e −021.49123.0357e −042.48583.1616e −063.48103.1718e −084.418712809.0650e −031.49605.3912e −052.49332.8666e −073.46321.2685e −081.3222Table 2
Errors and convergence rate of the Chebyshev point with differentd.
λ=2nd=2d=3d=4d=5ErrorhαErrorhαErrorhαErrorhα102.4329e + 001.3897e + 002.5216e −011.1033e −01208.8604e −011.45721.3680e −026.66654.5076e −035.80581.5195e −036.1821401.8242e −012.28012.8639e −032.25611.4419e −044.96647.2659e −067.7082803.2188e −022.50261.7517e −044.03113.0543e −065.56101.0125e −076.16521605.2215e −032.62408.5892e −064.35015.9098e −085.69161.0071e −06—3208.0770e −042.69262.9760e −074.85111.9760e −06—2.2948e −05—6401.2077e −042.74162.8632e −06—6.856e −05—1.0211e −03—12801.7925e −052.75227.4861e −05—1.2910e −02—9.6669e −02—In Table1, the convergence rate of equidistant nodes with different d is Ohd−2; in Table 2, the convergence rate of the Chebyshev point of the second kind with different d is Ohd+2,d≥2.For different values ofλ and different number of nodes, we can calculate the corresponding relative error; some of the data are shown in Tables 3 and 4.Table 3
Errors and convergence rate of the equidistant nodes with differentλ.
d=4nλ=−5λ=1λ=5λ=20ErrorhαErrorhαErrorhαErrorhα103.6809e + 012.2939e −012.1508e + 023.0238e + 08201.3259e + 011.47312.6894e −023.09256.1838e + 011.79838.0305e + 08—402.2319e + 002.57072.7208e −033.30529.5448e + 002.69576.7632e + 082.4778e −01802.6875e −013.05392.5675e −043.40561.1054e + 003.11011.6623e + 082.02461602.7599e −023.28352.3416e −053.45481.1147e −013.30982.4028e + 072.79043202.6249e −033.39432.1009e −063.47841.0509e −023.40702.7001e + 063.1536Table 4
Errors and convergence rate of the Chebyshev point with differentλ.
d=4nλ=−5λ=1λ=5λ=20ErrorhαErrorhαErrorhαErrorhα101.0975e + 011.8992e −024.4245e + 011.3431e_01201.5012e −016.19192.6112e −046.18451.3603e + 005.02352.5740e + 082.3835405.5289e −034.76308.0762e −065.01494.7712e −024.83341.4800e + 074.1204801.1729e −045.55891.6777e −075.58911.0350e −035.52673.5715e + 055.37291602.2528e −065.70221.8130e −083.21002.3528e −055.45918.3139e + 035.42493201.3043e −05—2.5236e −06—2.4682e −063.25292.1982e + 025.2411From Tables3 and 4, we can find that, for different values of λ, the convergence rate can reach Ohd+2d≥2 both for equidistant and nonequidistant nodes.


## 5. Conclusion

In this paper, the barycentric rational collocation method for solving third-order two-point boundary value equations is presented, and the error function of the convergence rateOhd−2 is also obtained. For the constant coefficient and variable coefficient of two-point boundary value equations, numerical results show that the convergence rate can reach Ohd−2 for the equidistant nodes and Chebyshev point of the second kind with d≥2, so the barycentric rational collocation method is an effective method. Compared with other methods, the advantage of this method is that the matrix equation can be easily obtained, the program is simple, and high computational accuracy can be obtained by using a few points.



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