# Barycentric Rational Collocation Method for Nonlinear Heat Conduction Equation

Authors: Jin Li
Journal: Journal of Applied Mathematics
 (2022)

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## Abstract

Nonlinear heat equation solved by the barycentric rational collocation method (BRCM) is presented. Direct linearization method and Newton linearization method are presented to transform the nonlinear heat conduction equation into linear equations. The matrix form of nonlinear heat conduction equation is also obtained. Several numerical examples are provided to valid our schemes.


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## Body

## 1. Introduction

In this article, the nonlinear heat conduction equation is considered as(1)∂ux,t∂t=cx,t,u∂2ux,t∂x2+Fx,t,u,∂ux,t∂x,a≤x≤b,t≥0,whereFx,t,u,∂ux,t/∂x is the nonlinear term of u and ux, with initial condition
(2)ux,0=fx,a≤x≤b,and the boundary condition
(3)ua,t=φ1tub,t=φ2t,t≥0.Combining (1)–(3), we get nonlinear heat conduction equation.The linear and nonlinear heat conduction equation is a second-order linear partial differential equation [1–6]. There are analytical and numerical methods to solve the linear/nonlinearheat conduction equation. For the numerical methods, there are the finite difference method, finite element method and boundary element method, the spectral method [7], etc.The barycentric formula have been obtained by the Lagrange interpolation formula [8–10] and been used to solve Volterra equation and Volterra integro-differential equation [11–14]. Floater and Kai, Klein and Berrut [15–17] have proposed a rational interpolation scheme and get the equidistant node of the barycentric formula. In recent papers, references [18–25] have been extended the barycentric collocation methods to solve initial/boudary value problems and linear/nonlinear problems.In this paper, the linear barycentric rational collocation method (LBRCM) for nonlinear heat conduction equation is presented. Direct linearization and Newton linearization are presented to transform the nonlinear heat conduction equation into linear equations. The matrix form of linearization scheme for nonlinear heat conduction equation is also obtained.This paper is organized as follows: in Section2, the linearization scheme for nonlinear heat conduction equation is given. Direct linearization and Newton linearization are constructed, and then the matrix form of LBRCM is also presented. At last, two numerical examples are listed to illustrate our numerical scheme.


## 2. Linearization

Direct linearization and Newton linearization are presented to change the nonlinear heat conduction equation into linear equation. By choosing the equidistant nodes and Chebyshev nodes as collocation point to solve the nonlinear heat conduction equation, the direct linearization scheme can be adopted as(4)∂ux,t∂t=cx,t,u0∂2ux,t∂x2+Fx,t,u0,∂u0x,t∂x,∂unx,t∂t=cx,t,un−1∂2unx,t∂x2+Fx,t,un−1,∂un−1x,t∂x.Newton linearization scheme can be adopted as(5)∂ux,t∂t=cx,t,u∂2ux,t∂x2+Fx,t,u0,∂u0x,t∂xu−u0+⋯,and then we get the Newton linearization as
(6)∂ux,t∂t=cx,t,u∂2ux,t∂x2+Fx,t,u0,∂u0x,t∂xu−u0,∂unx,t∂t=cx,t,un−1∂2unx,t∂x2+Fx,t,un−1,∂un−1x,t∂xun−un−1.We partition the intervala,b into a=x0<x1<⋯<xm=b,h=b−a/m and 0,T into 0=t0<t1<⋯<tn=T,τ=T/n with Ω=a,b×0,T and xi,tj,i=0,1,2,⋯,m;j=0,1,2,⋯,n. Then, we have
(7)rm,nx,t=∑i=0m∑j=0nrixrjtui,j,where ui,j=uxi,tj and
(8)rix=wi/x−xi∑j=0nwj/x−xj,rjt=wj/t−tj∑j=0mwj/t−tj,wk=∑i∈Jk−1i∏j=i,j≠ki+d1xk−xj,and Jk=i∈I;k−d≤i≤k,I=0,1,⋯,n−d.Taking equation (7) into equation (1), we can get the LBRCM as
(9)∑i=0m∑j=0nrixrj′tui,j=cx,t,∑i=0m∑j=0nrixrjtui,j∑i=0m∑j=0nr′′ixrjtui,j+Fx,t,∑i=0m∑j=0nrixrjtui,j,∑i=0m∑j=0nri′xrjtui,j.


## 3. Numerical Examples

In the following, two examples are given to valid our algorithm.Example 1.
Consider(10)∂u∂t=∂2u∂x2+e−u+e−2u,a<x<b;t≥0,with
(11)u0,t=lnt+2,u1,t=lnt+3,ux,0=lnx+2,
Its analysis solutions is(12)ux,t=lnx+t+2.
Firstly, direct linearization of nonlinear heat conduction equation is given as(13)∂u∂t=∂2u∂x2+e−u0+e−2u0,a<x<b;t>0,with its direct linearization scheme is
(14)∂un∂t=∂2un∂x2+e−un−1+e−2un−1,n=1,2,⋯,and its matrix form is
(15)D0,1−D2,0un=e−un−1+e−2un−1,n=1,2,⋯,where
(16)Dijm0=Rimxj,m=1,2,Dij01=Ri1tj,(17)Dij10=wi/wjxj−xi,j≠i,−∑i≠jDji10,j=i,Dji20=2Dji10Dji10−1xj−xi,j≠i,−∑i≠jDji20,j=i,(18)Dij01=wi/wjtj−ti,j≠i,−∑i≠jDji01,j=i.In Figure1, the analysis solution ux,t=lnx+t+2 is presented. In the following Figures 2 and 3, errors of equidistant nodes and Chebyshev nodes for LBRCM with m=n=10,d1=d2=7 are given, respectively; from the figure, we know that the accuracy of Chebyshev nodes is higher than equidistant nodes.Figure 1
Analysis solutions ofux,t=lnx+t+2.Figure 2
Errors of equidistant nodes for LBRCM withm=n=10,d1=d2=7.Figure 3
Errors of Chebyshev nodes for LBRCM withm=n=10,d1=d2=7.In the following Figures4 and 5, errors of equidistant nodes and Chebyshev nodes for BLCM with m=n=10 are given, respectively; from the figure, we know that the accuracy of Chebyshev nodes is also higher than equidistant nodes.Figure 4
Errors of equidistant nodes for BLCM withm=n=10.Figure 5
Errors of Chebyshev nodes for BLCM withm=n=10.In Tables1 and 2, the errors of direct linearization scheme by LBRCM with equidistant nodes for x and t are presented.Table 1
Errors of the LBRCM withd2=7 at t=1 for equidistant nodes.
m,nd1=1hαd1=2hαd1=3hαd1=4hα8, 85.8782e-041.4718e-056.2224e-065.3762e-0716, 161.8139e-041.69631.9147e-062.94243.6861e-074.07732.1906e-084.617232, 325.4855e-051.72542.5010e-072.93652.3239e-083.98747.8080e-104.810264, 641.6239e-051.75613.2280e-082.95381.5064e-093.94748.2417e-113.2439Table 2
Errors of the LBRCM withd1=7 at t=1 for equidistant nodes.
m,nd2=1ταd2=2ταd2=3ταd2=4τα8, 89.5811e-041.7102e-055.4941e-063.4075e-0716, 165.2156e-040.87745.2640e-061.70008.0961e-072.76263.2948e-083.370432, 322.3682e-041.13901.3002e-062.01759.8320e-083.04172.2777e-093.854564, 649.5433e-051.31122.8040e-072.21321.0531e-083.22281.2354e-104.2045In Tables3 and 4, the errors of the LBRCM with Chebyshev nodes for x and t are presented.Table 3
Errors of the LBRCM withd2=7 at t=1 for Chebyshev nodes.
m,nd1=1hαd1=2hαd1=3hαd1=4hα8, 81.3717e-031.5290e-056.3486e-061.6112e-0716, 163.3177e-042.04771.2090e-063.66078.3105e-086.25543.2477e-095.632632, 327.7738e-052.09357.4116e-084.02792.5853e-095.00651.5222e-09—64, 648.4472e-05—4.3077e-06—5.6046e-06—2.0141e-06—Table 4
Errors of the LBRCM withd1=7 at t=1 for Chebyshev nodes.
m,nd2=1ταd2=2ταd2=3ταd2=4τα8, 85.7527e-048.8997e-061.7666e-068.3203e-0816, 161.4898e-041.94921.2899e-062.78651.5065e-073.55175.5006e-093.919032, 323.5408e-052.07291.5959e-073.01489.1369e-094.04332.7123e-104.342064, 648.7961e-062.00912.9251e-082.44781.2933e-07—1.2630e-07—Secondly, Newton linearization of nonlinear heat conduction equation is given as(19)∂u∂t=∂2u∂x2+e−u0+e−2u0u−u0,a<x<b;t≥0,with its Newton linearization scheme that is
(20)∂un∂t=∂2un∂x2+e−un−1+e−2un−1un−un−1,n=1,2,⋯,and its matrix form is
(21)D0,1−D2,0+diage−un−1+2e−2un−1un=e−un−1+e−2un−1+e−un−1+2e−2un−1∘un−1,n=1,2,⋯,where D0,1 and D2,0 are defined as (16).In Tables5 and 6, the errors of Newton linearization scheme by LBRCM with equidistant nodes for x and t are presented.Table 5
Errors of the LBRCM withd2=7 at t=1 for equidistant nodes.
m,nd1=1ταd1=2hαd1=3hαd1=4hα8, 89.5811e-041.7102e-055.4941e-063.4075e-0716, 165.2156e-040.87745.2640e-061.70008.0961e-072.76263.2949e-083.370432, 322.3682e-041.13901.3002e-062.01759.8318e-083.04172.2716e-093.858464, 649.5433e-051.31122.8038e-072.21321.0422e-083.23781.8173e-103.6439Table 6
Errors of the LBRCM withd1=7 at t=1 for equidistant nodes.
m,nd2=1ταd2=2ταd2=3ταd2=4τα8, 85.8782e-041.4718e-056.2224e-065.3762e-0716, 161.8139e-041.69631.9147e-062.94243.6861e-074.07732.1906e-084.617232, 325.4855e-051.72542.5009e-072.93662.3236e-083.98777.7787e-104.815764, 641.6239e-051.75613.2275e-082.95401.4867e-093.96615.7909e-113.7477In Tables7 and 8, the errors of Newton linearization by LBRCM with Chebyshev nodes for x and t are presented.Table 7
Errors of the LBRCM withd2=7 at t=1 for Chebyshev nodes.
m,nd1=1hαd1=2hαd1=3hαd1=4hα8, 85.7527e-048.8997e-061.7666e-068.3204e-0816, 161.4898e-041.94921.2899e-062.78651.5065e-073.55175.5041e-093.918132, 323.5408e-052.07291.5957e-073.01508.8804e-094.08441.7617e-104.965464, 648.7936e-062.00962.4359e-082.71165.8007e-08—2.0160e-07—Table 8
Errors of the LBRCM withd1=7 at t=1 for Chebyshev nodes.
m,nd2=1ταd2=2ταd2=3ταd2=4τα8, 81.3717e-031.5290e-056.3486e-061.6112e-0716, 163.3177e-042.04771.2090e-063.66078.3105e-086.25543.2477e-095.632632, 327.7738e-052.09357.4196e-084.02632.2857e-095.18424.6534e-102.803164, 648.1984e-05—5.4975e-07—7.8881e-06—9.1447e-07—From the table above, we know that the accuracy of Newton linearization scheme is higher than direct linearization scheme.Example 2.
Consider(22)∂u∂t=α∂2u∂x2−u∂u∂x+fx,t,a<x<b;t>0,with the condition
(23)fx,t=sin6πx36π2cost−sint+6πcos6πxcos2t,ux,0=sin6πx,u0,t=0.
Its analysis solutions is(24)ux,t=costsin6πx.
In this example, direct linearization scheme is presented(25)∂u∂t=∂2u∂x2−u∂u∂x+e−u0+e−2u0,a<x<b;t>0,with its direct linearization scheme is
(26)∂un∂t=∂2un∂x2−un−1∂un∂x+e−un−1+e−2un−1,n=1,2,⋯,and its matrix form is
(27)D0,1−αD2,0+diagun−1D1,0un=F,n=1,2,⋯,where D0,1 and D2,0 are defined as (16).In Figure6, the analysis solution ux,t=costsin6πx is presented. In the following Figures 7 and 8, errors of equidistant nodes and Chebyshev nodes for LBRCM with m=n=31,d1=d2=15 are given, respectively; from the figure, we know that the accuracy of Chebyshev nodes is higher than equidistant nodes.Figure 6
Analysis solutions ofux,t=costsin6πx.Figure 7
Errors of equidistant nodes for LBRCM withm=n=31,d1=d2=15.Figure 8
Errors of Chebyshev nodes for LBRCM withm=n=31,d1=d2=15.In Tables9 and 10, the errors of direct linearization by LBRCM with equidistant nodes for x and t are presented.Table 9
Errors of the LBRCM withd2=7 at t=1 for equidistant nodes.
m,nd1=1hαd1=2hαd1=3hαd1=4hα8, 85.0725e-011.6979e+007.4870e+005.0517e+0116, 163.8989e-023.70161.3298e-013.67451.0527e-016.15233.6951e-0210.41732, 328.2371e-032.24281.4130e-023.23441.9625e-035.74523.7448e-033.302764, 642.3338e-031.81941.6289e-033.11684.9679e-068.62591.2120e-044.9495Table 10
Errors of the LBRCM withd1=7 at t=1 for equidistant nodes.
m,nd2=2ταd2=3ταd2=4τα8, 83.0358e+013.5536e+013.2379e+013.1088e+0116, 164.6120e-029.36254.5581e-029.60664.5403e-029.47814.5327e-029.421832, 327.1933e-046.00267.0368e-046.01747.0268e-046.01387.0221e-046.012364, 642.0118e-055.16019.7053e-079.50199.6556e-079.50739.6539e-079.5066In Tables11 and 12, the errors of direct linearization by LBRCM with Chebyshev nodes for x and t are presented.Table 11
Errors of the LBRCM withd2=7 at t=1 for Chebyshev nodes.
m,nd1=1hαd1=2hαd1=3hαd1=4hα8, 83.8779e+003.7472e+003.6747e+003.6317e+0016, 167.1754e-039.07807.0233e-039.05947.0547e-039.02486.9460e-039.030332, 323.3421e-057.74627.5693e-069.85787.9371e-069.79578.8251e-069.620364, 641.3473e-051.31074.5503e-087.37804.4917e-0910.7878.2779e-0910.058Table 12
Errors of the LBRCM withd1=7 at t=1 for Chebyshev nodes.
m,nd2=1ταd2=2ταd2=3ταd2=4τα8, 84.5490e+002.1362e+002.1145e+002.6899e+0016, 167.7786e-025.86993.0474e-026.13131.5449e-027.09679.2472e-038.184332, 321.6150e-022.26792.5803e-033.56205.0884e-044.92412.0372e-045.504364, 641.1859e-020.44561.1541e-031.16072.1401e-041.24954.4209e-052.2042


## 4. Concluding Remarks

The nonlinear heat conduction equation is solved by LBRCM with direct linearization scheme and Newton linearization scheme, different from the finite difference methods with the time variable computed separately, while the LBRCM can get the space variable and time variable simultaneously. The matrix form of LBRCM with direct linearization scheme and Newton linearization scheme is given from the corresponding scheme, and numerical results are given to valid our scheme. In the table, the convergence of LBRCM for equidistant nodes and Chebyshev nodes has been given, while the theorem analysis of direct linearization scheme and Newton linearization scheme is out of goal in this paper which will be given in the near future.



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