# Linear Barycentric Rational Method for Two-Point Boundary Value Equations

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

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## Abstract

Linear barycentric rational method for solving two-point boundary value equations is presented. The matrix form of the collocation method is also obtained. With the help of the convergence rate of the interpolation, the convergence rate of linear barycentric rational collocation method for solving two-point boundary value problems is proved. Several numerical examples are provided to validate the theoretical analysis.


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## Body

## 1. Introduction

The analysis of many physical phenomena and engineering problems can be reduced to solving the boundary value problem of differential equation, most of which need to be solved by the numerical method. The barycentric interpolation method is a high precision calculation method, and a strong form of collocation that relies on differential equation, which has been studied extensively by many scholars. The linear barycentric rational method (LBRM) [1–3] has been used to solve certain problems such as delay Volterra integro-differential equations [4], Volterra integral equations [5–7], biharmonic equation [8], beam force vibration equation [9], boundary value problems [10], heat conduction problems [11], plane elastic problems [12], incompressible plane elastic problems [13], nonlinear problems [14], and so on [1, 15].In this article, we pay our attention to the numerical solution of two-point boundary value problems:(1)Tux≔u″x+qux=fx,x∈a,b,(2)ua=uℓ,ub=ur.Let the intervala,b be partitioned into n uniform part with h=b−a/n and x0,x1,…,xn with its related function fxi,i=0,1,…,n. For any 0≤d≤n, with Pxi,i=0,1,…,n−d, to be 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.Change the polynomialPix into the Lagrange interpolation form as(5)Pix=∑k=ii+d∏j=i,j≠ki+dx−xjxk−xjfk.Combining (7) and (5) together, we get(6)∑i=0n−dλixPix=∑i=0n−d−1i∑k=ii+d1x−xk∏i,j≠ki+d1xk−xjfk=∑k=0nwkx−xkfk,where wk=∑i∈Jk−1i∏j=i,j≠ki+d1/xk−xj and 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, its weight function is(9)wj=−1n−jCnj.The Chebyshev point of the second kind is(10)xj=cosjπn,j=0,1,…,n,and its weight function is(11)wj=−1jδj,δj=12,j=0,n,1,ortherwise.Consider the barycentric interpolation function as(12)unx=∑j=0nLjxuj,and the numerical scheme is given as(13)∑j=0nujLj″x+q∑j=0nujLjx=fx.By using the notation of the differential matrix, equation (13) is denoted as matrices in the form of(14)∑j=0nDij2uj+q∑j=0nδijuj=fxi,where i=1,2,…,n.Equation (13) is written as matrices in the form of(15)D2+qIu=f,where L≔D2+qI,u=u0,u1,u2,…,unT,Dk=Dijkn+1×n+1,(16)Dij1=ωj/ωixi−xj,i≠j,−∑k≠iDik1,i=j,Dij2=2Dij1Dii1−1xi−xj,i≠j,−∑k≠iDik2,i=j,and q=diagq,f=fx0,fx1,fx2,…,fxnT. Using interpolation formulas, boundary conditions can be discretized into(17)∑j=0nD1j1uj=a,∑j=0nDnj1uj=b.


## 2. Convergence and Error Analysis

With the error function of difference formula(18)ex≔ux−rx=x−xi⋯x−xi+dxi,xi+1,…,xi+d,xf,and(19)ex=∑i=0n−dλixux−Pix∑i=0n−dλix=AxBx=Ohd+1,where Ax≔∑i=0n−d−1ixi,…,xi+d,xf,Bx≔∑i=0n−dλix. Taking the numerical scheme(20)∑j=0nyjLj″x+q∑j=0nyjLjx=fx.Combining (20) and (1), we have(21)Tex≔e″x+qex=Rfx,where Rfx=fx−fxk,k=0,1,2,…,n.The following Lemma has been proved by Jean-Paul Berrut in [13].Lemma 1 (see [13]).
Forex defined in (18), we have(22)ex≤Chd+1,u∈Cd+2a,b,e′x≤Chd,u∈Cd+3a,b,e″x≤Chd−1,u∈Cd+4a,b,d≥1.
Letux be the solution of (1) and unx is the numerical solution, then we have(23)Tunxk=fxk,k=0,1,2,…,n,and(24)limn⟶∞unx=ux.The results can be obtained in the reference of [14].Based on the above lemma, we derive the following theorem.Theorem 1.
Letunx:Tunx=fx,un∗x:Tun∗x=f∗x, and fx∈Ca,b, we have(25)unx−un∗x≤Chd−1.Proof.
AsL≔D2+QI, where(26)D2=D002D012D022D032⋯D0n2D102D112D122D132⋯D1n2D202D212D222D232⋯D2n2D302D312D322D332⋯D3n2………………Dn−1,02Dn−1,12Dn−1,22Dn−2,32⋯Dn−1,n2Dn02Dn12Dn22Dn32⋯Dnn2,and(27)L≔D2+QI=q+D002D012D022D032⋯D0n2D102q+D112D122D132⋯D1n2D202D212q+D222D232⋯D2n2D302D312D322D332⋯D3n2………………Dn−1,02Dn−1,12Dn−1,22Dn−2,32⋯Dn−1,n2Dn02Dn12Dn22Dn32⋯q+Dnn2.
Putting column 2, column 3, columnn added to column 1, we have(28)D2=0D012D022D032⋯D0n20D112D122D132⋯D1n20D212D222D232⋯D2n20D312D322D332⋯D3n20……………0Dn−1,12Dn−1,22Dn−2,32⋯Dn−1,n20Dn12Dn22Dn32⋯Dnn2,which means the matrix D2 is the singular matrix.
Similarly we have(29)L=qD012D022D032⋯D0n2qq+D112D122D132⋯D1n2qD212q+D222D232⋯D2n2qD312D322q+D332⋯D3n2q……………qDn−1,12Dn−1,22Dn−2,32⋯Dn−1,n2qDn12Dn22Dn32⋯q+Dnn2,and then we assume L≠0 with q≠0, unx=∑j=0nLjxfj,un∗x=∑j=0nLjxfj∗, where. Un=fx0,fx1,…,fxnTandUn∗=f∗x0,f∗x1,…,f∗xnT.
By(30)Un−Un∗=L−1LUn−Fn∗,which means(31)unx−un∗x=∑j=0nMjxTex,where Mjx is the element of matrix L−1.
Then we have(32)unx−un∗x≤∑j=0nMjxTex≤Chd−1.
The proof is completed.
We know that the central difference method can achieve quadratic convergence and the convergence order is the same as that ofd = 3. When d >3, the convergence of the barycentric rational method is better than that of the central difference method.


## 3. Numerical Example

Example 1.
Consider the two-point boundary value:(33)−y″+400y=−400cos2πx−2π2cos2πx,(34)y0=y1=0,and its analysis solution is(35)yx=e−201+e−20e20x+11+e−20e−20x−cos2πx.
In this example, we consider the two-point boundary value equations with the boundary conditiony0=y1=0. In Table 1, the convergence rate of equidistant nodes with different d is Ohd; in Table 2, the convergence rate of the Chebyshev point of the second kind with different d is Ohd+2,d≥2. From Theorem 1, the convergence rate is Ohd−1, and there are no convergence rates as d=1. Here the convergence rate is Oh and Oh2 in Tables 1 and 2 for d=1, respectively, and we will give exact analysis in other paper.Table 1
Errors of the equidistant nodes with differentd.
nd=1d=2d=3d=4d=5105.6796e − 024.7174e − 023.6770e − 022.9692e − 022.5240e − 02203.1866e − 022.0707e − 021.2295e − 027.5473e − 034.6949e − 03401.1899e − 024.8530e − 031.8024e − 036.8958e − 042.6519e − 04804.0465e − 038.2037e − 041.7027e − 043.6416e − 057.8060e − 061601.4721e − 031.1821e − 041.2993e − 051.4706e − 061.6657e − 073204.9182e − 041.5796e − 058.9500e − 075.2131e − 083.0430e − 096401.5719e − 042.0375e − 065.8674e − 081.7409e − 095.1202e − 11Table 2
Errors of the Chebyshev point with differentd.
nd=1d=2d=3d=4d=5104.7235e − 023.4143e − 022.3198e − 021.0658e − 024.2299e − 03202.0966e − 024.9675e − 032.3057e − 031.5490e − 031.0185e − 03405.0812e − 033.3376e − 047.5530e − 053.1018e − 051.2517e − 05801.2482e − 032.0099e − 052.1222e − 064.5138e − 079.5181e − 081602.9995e − 041.2091e − 066.1512e − 086.5988e − 096.9630e − 103207.3360e − 057.3634e − 081.8414e − 091.0610e − 102.8788e − 106401.8092e − 054.5328e − 095.7970e − 111.5463e − 106.0430e − 09Example 2.
Consider the two-point boundary value.(36)y″+y′sinx+yex=−16π2sin4πx+4πsinxcos4πx+ex2+sin4πx,−1<x<1,with the boundary condition(37)y−1+y′−1=2+4π,y1+y′1=2+4π,and its analysis solution is(38)yx=2+sin4πx.
In this example, we consider the variable coefficient of two-point boundary value equations with the boundary conditiony−1+y′−1=2+4π,y1+y′1=2+4π. In Table 3, the convergence rate of equidistant nodes with different d is Ohd; in Table 4, the convergence rate of the Chebyshev point of second kind with different d is Ohd+2,d≥2.Table 3
Errors of the equidistant nodes with differentd.
nd=1d=2d=3d=4d=5103.1903e + 005.8183e + 004.2584e + 003.3515e + 003.8150e + 00209.0854e − 011.8487e − 014.7472e − 025.6182e − 022.1227e − 02401.9690e − 013.8711e − 025.9273e − 041.9234e − 044.9300e − 05804.4235e − 023.9481e − 034.9399e − 062.3577e − 062.4228e − 071609.6887e − 033.4240e − 041.9523e − 071.9939e − 082.8181e − 093202.0060e − 032.7549e − 055.5624e − 091.6331e − 097.9052e − 086403.7877e − 042.1177e − 061.0976e − 092.1374e − 081.2137e − 06Table 4
Errors of the Chebyshev point with differentd.
nd=1d=2d=3d=4d=5101.3687e + 011.0866e + 022.9315e + 026.7980e + 021.4700e + 03202.9375e + 002.2412e + 012.0846e + 012.2744e + 003.1001e + 01401.0678e + 004.9833e + 009.5860e − 011.3524e + 007.5788e − 01804.4216e − 011.1571e + 002.6706e − 029.3346e − 028.5273e − 031601.9821e − 012.7788e − 011.7104e − 035.6811e − 032.5039e − 053209.3181e − 026.8032e − 024.9655e − 043.4460e − 042.6914e − 066404.5032e − 021.6827e − 027.8365e − 052.1132e − 051.3480e − 07


## 4. Concluding Remarks

In this paper, the numerical approximation of linear barycentric rational collocation method for solving two-point boundary value equations is presented. The matrix form of the algorithm is given for the simple calculation; with the help of Newton formula, the error function of the convergence rateOhd−1 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 for the equidistant nodes and Ohd+2 for the Chebyshev point of the second kind with d≥2. For the special case of d=1, there are still convergence rates with Oh, and the analysis of this phenomenon will be presented in other papers.



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