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### formerly Central European Journal of Physics

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Volume 16, Issue 1

# On the bivariate spectral quasi-linearization method for solving the two-dimensional Bratu problem

Hillary Muzara
/ Stanford Shateyi
• Corresponding author
• Department of Mathematics and Applied Mathematics, University of Venda, Thohoyandou 0950, South Africa
• Email
• Other articles by this author:
/ Gerald Tendayi Marewo
Published Online: 2018-08-20 | DOI: https://doi.org/10.1515/phys-2018-0072

## Abstract

In this paper, a bivariate spectral quasi-linearization method is used to solve the highly non-linear two dimensional Bratu problem. The two dimensional Bratu problem is also solved using the Chebyshev spectral collocation method which uses Kronecker tensor products. The bivariate spectral quasi-linearization method and Chebyshev spectral collocation method solutions converge to the lower branch solution. The results obtained using the bivariate spectral quasi-linearization method were compared with results from finite differences method, the weighted residual method and the homotopy analysis method in literature. Tables and graphs generated to present the results obtained show a close agreement with known results from literature.

## 1 Introduction

The most of real life phenomena are modeled by partial differential equations (PDEs). In science, engineering, biological sciences and fluid mechanics, most of these phenomena are described by PDEs, which are usually non-linear [1, 2]. Due to the complexity of the domains in which they are defined, it is usually very difficult or even impossible, except for a few special cases, to find exact solutions to the defining PDEs. This motivated researchers to develop numerical and analytical methods to approximate solutions to these non-linear PDEs. Some of the well known analytical methods that have been used to solve non-linear PDEs include homotopy pertubation method [3], Adomain decomposition method [4], power series expansions [5], the artificial small parameter method [6] and the δ-pertubation expansion method [7]. Although these methods help us to understand many non-linear phenomena, they have their own disadvantage in that the convergence of the solution series is not guaranteed due to their dependence on small or large physical parameters. Some examples of numerical methods that have been used to solve non-linear phenomena include finite element methods [8], finite difference methods [9], quasi-linearization technique [10], iterative finite difference method [11], the B-spline method [12]. It is worth mentioning that the fractional-wavelet approach is an important technique which can be used to solve nonlinear PDEs. For further reading on the fractional-wavelet approach and fractional calculus, we refer interested readers to [13,14].

An example of a highly non-linear differential equation is the so-called Bratu problem, first set up by Bratu and named after him [15]. The simplest form of the Bratu problem in one dimension is:

$d2udx2+λeu(x)=0,x∈[0,1]$(1)

subject to boundary conditions u(0) = u(1) = 0. The exact solution of Eq. (1) is given by

$u(x)=−2ln⁡[cosh⁡((x−12)ω2)cosh⁡(ω4)]$(2)

where ω is the solution of the equation

$\begin{array}{}\omega =2\sqrt{2\lambda }\mathrm{cosh}\left(\frac{\omega }{4}\right)\end{array}$[16]. The Bratu problem has no, unique, or two solutions if λ > λc, λ = λc, or λ < λc respectively, where the critical value λc = 3.51382 [17]. The generalization of the Bratu problem is the Liouville-Bratu-Gelfand problem [16] which in the n-dimensional coordinate system takes the form

$Δu(x)+λeu(x)=0,x∈Ω$(3)

where the square domain Ω is bounded in ℝn together with homogeneous Dirichlet boundary conditions u(x) = 0, x∂ Ω, where ∂ Ω is the boundary of Ω.

In this study, we consider the two dimensional Bratu problem, which has the form

$∂2u∂x2+∂2u∂y2+λeu=0,(x,y)∈[0,1]×[0,1]$(4)

subject to boundary conditions

$u(0,y)=u(1,y)=u(x,0)=u(x,1)=0$(5)

where λ is a positive number. Similar to the 1D case, depending on the value of the parameter λ, Eq. (4) has no, one, or two solutions if λ > λc, λ = λc or λ < λ, respectively, where the critical value λc = 6.808124423 [18]. The two dimensional Bratu problem has no exact solution as is the case with the one dimensional Bratu problem. However, Odejide and Aregbesola [19] presented a near exact solution that satisfies Eq. (4) at just one collocation point (xc,yc) = ($\begin{array}{}\frac{1}{2},\frac{1}{2}\end{array}$) as well as the boundary conditions. The analytical solution is given as

$u(x,y)=2ln⁡[cosh⁡(ω4)cosh⁡[(x−12)(y−12)ω]cosh⁡((x−12)ω2)cosh⁡(y−12)ω2]$(6)

where ω is a solution of the equation

$ω2=λcosh2⁡(ω4).$(7)

The two dimensional Bratu problem has attracted the attention of many researchers because of its wide range of physical, chemical and engineering applications. It is reported in [16] that the Bratu problem is used to model the thermal reaction process in a combustible non-deformable material. The Bratu problem also appears in the Chandrasekhar model of the expansion of the universe, chemical reactor theory and nanotechnology [20]. Recently, the Bratu problem has found applications in engineering such as electro-spinning process for the fabrication of nano-fibers [21]. Apart from the physical applications, the Bratu problem is also used as a benchmark for newly developed numerical and analytical methods [22].

Some of the numerical methods that have been used to treat the Bratu problem in two dimensions include: a wavelet homotopy analysis method (wHAM) by Zhaochen and Shijun [22], the finite difference (FD) and the weighted residual method (WRM) by Odejide and Aregbesola [19] and the Chebyshev pseudospectral method using Gegenbauer polynomials [23] by Boyd.

The main objective of this work is to solve (for the first time), the so-called Bratu problem in two dimensions using the bivariate spectral quasi-linearization method (BSQLM). The BSQLM introduced by Motsa et al [24] is a modification of the spectral quasi-linearization method [25] to solve non-linear PDEs in two dimensions. Some of the problems successfully solved using the BSQLM include the modified Kdv equation, Burger equation, the Cahn-Hillard equation and the Fitzhugh-Nagumo equations [24]. It is in this work that researchers concluded that the method is accurate, reliable and applicable to nonlinear evolution equations. The obtained results also showed that the method achieves high accuracy with relatively fewer spatial grid points and converges fast to the exact solution. Also, Motsa and Ansari [26] solved non-dimensionalized PDEs describing a time dependent boundary layer flow and heat transfer of an incompressible Oldroyd-B nanofluid past an impulsively stretching sheet using the BSQLM. The results obtained converged rapidly.

## 2.1 Bivariate spectral quasi-linearization method

In this section, we briefly describe the Bivariate spectral quasi-linearization method (BSQLM). The BSQLM uses the quasi-linearization method, Chebyshev collocation method and bivariate Lagrange interpolation.

## 2.1.1 Quasi-linearization method

The quasi-linearization method (QLM) which is based on Newton-Raphson method was introduced by Bellman and Kalaba [27]. It is a technique for simplifying non-linear PDEs using the linear terms of the Taylor series expansion about an initial approximation. There are some techniques, which can be used to transform nonlinear PDEs to linear equations. As an example, the Hopf-Cole transformation discussed in [28, 29], is used to turn a strongly nonlinear Burgers equation into a linear equation. In this work, let us consider a general second order non-linear differential equation

$F(u)=R(x),x=(x,y)∈[a,b]×[c,d]$(8)

subject to given boundary conditions. The unknown function $\begin{array}{}\mathbf{u}=\left(u,\frac{\mathrm{\partial }u}{\mathrm{\partial }x},\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{x}^{2}},\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }x\mathrm{\partial }y},\frac{\mathrm{\partial }u}{\mathrm{\partial }y},\frac{{\mathrm{\partial }}^{2}u}{\mathrm{\partial }{y}^{2}}\right)\end{array}$ and F is a non-linear operator. Expanding F using linear Taylor series expansion about v we get

$F(v)+∇F(v)⋅(u−v)≈R(x)$(9)

Assuming that v is an approximate solution sufficiently close to u and adopting the notation ur and ur+1 for v and u respectively, we have

$∇F(ur)⋅ur+1=∇F(ur)⋅ur−F(ur)+R(x)$(10)

where r = 0, 1, 2, ⋯. Solving Eq. (10) generates a sequence {ur} and hence ur such that uru as r → ∞.

## 2.1.2 Bivariate Lagrange interpolation and Chebyshev differentiation

We seek the solution u of Eq. (10) of the form

$u(x,y)≈∑j=0N∑i=0Mu(xj,yi)Lji(x,y)$(11)

where the functions Lji(x,y) = Lj(x)Li(y) are bivariate Lagrange polynomials defined as

$Lj(x)=∏k=0k≠jNx−xkxj−xk,Li(y)=∏k=0k≠iMy−ykyi−xk$(12)

The functions Lj(x) and Li(y) both obey the Kronecker delta equation, that is,

$Lji(xn,ym)=1,ifj=n,i=m0,otherwise$(13)

Before applying the spectral method, it is convenient to transform the physical domain [a,b] × [c,d] in the x-y axis to the computational domain [–1, 1] × [–1, 1] in the ξ-η axis using linear transformations x(ξ) = $\begin{array}{}\frac{a+b}{2}+\frac{b-a}{2}\end{array}$ ξ and y(η) = $\begin{array}{}\frac{c+d}{2}+\frac{d-c}{2}\eta .\end{array}$ Approximating the partial derivatives of u at Chebyshev-Gauss-Lobatto collocation points

${xj}j=0N=cos⁡(πjN),{yi}i=0M=cos⁡(πiM),$(14)

we have

$∂u∂x|x=xjy=yi≈∑p=0N∑q=0Mu(xp,yq)dLp(xj)dxLq(yi)$(15)

$=∑p=0NDjpu(xp,yi)=DUi.$(16)

$∂u∂y|x=xjy=yi≈∑q=0Mdiqu(xj,yq)=∑q=0MdiqUq,$(17)

and

$∂2u∂x∂y|x=xjy=yi≈∑p=0N∑q=0Mu(xp,yq)dLp(xj)dxdLq(yi)dy=∑q=0Mdiq(DUq),$(18)

where = $\begin{array}{}\left(\frac{b-a}{2}\right)\end{array}$ and iq= $\begin{array}{}\left(\frac{d-c}{2}\right)\end{array}$diq are the standard Chebyshev differentiation matrices [30] of orders (N + 1) × (N + 1) and (M + 1) × (M + 1), respectively, and Ui = (u(x0,yi),u(x1,yi), ⋯, u(xN,yi))T, for i = 0, 1, 2, ⋯, N. The superscript T denotes matrix transposition. Higher order derivatives of u are defined as follows:

$∂nu∂xn|x=xjy=yi≈DnUi,∂nu∂yn|x=xjy=yi≈∑q=0MdiqnUq,n=2,3,⋯.$

and

$∂n+mu∂xn∂ym=∑q=0Mdiqm(DnUq),n,m=1,2,⋯$

Expanding Eq. (10) we get

$α5,r∂2ur+1∂x2+α4,r∂2ur+1∂y2+α3,r∂2ur+1∂x∂y+α2,r∂ur+1∂x+α1,r∂ur+1∂y+α0,rur+1=Kr,$(19)

where

$α5,r=∂F(ur)∂uxx,α4,r=∂F(ur)∂uyy,α3,r=∂F(ur)∂uxy,α2,r=∂F(ur)∂ux,α1,r=∂F(ur)∂uy,α0,r=∂F(ur)∂u$

and

$Kr=α5,r∂2ur∂x2+α4,r∂2ur∂y2+α3,r∂2ur∂x∂y+α2,r∂ur∂x+α1,r∂ur∂y+α0,rur−F(ur)+R(x).$

Approximating the u and its derivatives in Eq. (19) at collocation points (xj,yi) we get

$α5,r(x,yi)D^2Ur+1,i+α4,r(x,yi)∑q=0Md^jq2Ur+1,q+α3,r(x,yi)∑q=0Md^iq2(D^2Ur+1,q)+α2,r(x,yi)D^Ur+1,i+α1,r(x,yi)∑q=0Md^iq(D^Ur+1,q)+α0,i(x,yi)Ur+1,i=Kr,i$(20)

where

$αa,r(X,yi)=αa,r(x0,yi)αa,r(x1,yi)⋱αa,r(xN,yi)$

and

$Kr,i=α5,r(x,yi)D^2Ur,i+α4,r(x,yi)∑q=0Md^jq2Ur,q+α3,r(x,yi)∑q=0Md^iq2(D^2Ur,q)+α2,r(x,yi)D^Ur,i+α1,r(x,yi)∑q=0Md^iq(D^Ur,q)+α0,i(x,yi)Ur,i−F(ur(x,yi))+R(x,yi).$

In compact form, Eq. (20) can be written as:

$A0,0A0,1⋯A0,MA1,0A1,1⋯A1,M⋮⋮⋱⋮AN−1,0AN−1,1⋯AN−1,MAN,0AN,1⋯AN,MUr+1,0Ur+1,1⋮Ur+1,N−1Ur+1,N=K0K1⋮KM−1KM$

where

$Aii=α5,r(x,yi)D^2+α4,r(x,yi)d^iiI+α3,r(x,yi)d^ii2D^2+α2,rx,yiD^+α1,r(x,yi)d^iiD^+α0,r(x,yi)Aij=d^ijI,wheni≠j$

## 2.2 Application to the current problem

The equivalent form of Eq. (4) in the ξ-η axis is

$∂2u∂ξ2+∂2u∂η2+λ4eu(ξ,η)=0,(ξ,η)∈[−1,1]×[−1,1]$(21)

subject to boundary conditions

$u(−1,η)=u(1,η)=u(ξ,−1)=u(ξ,1)=0.$(22)

Applying formula Eq. (10) on Eq. (21) we get its linear counterpart

$∂2ur+1∂ξ2+∂2ur+1∂η2+λ4eurur+1=λ4eur[ur−1],$(23)

r = 0, 1, 2, ⋯ together with boundary conditions

$ur+1(−1,η)=ur+1(1,η)=ur+1(ξ,−1)=ur+1(ξ,1)=0.$

Applying the spectral collocation method we get

$∑p=0ND^jp2Ur+1(ξp,ηi)+∑q=0Md^iq2Ur+1(ξj,ηq)+λ4eUr(ξj,ηi)Ur+1(ξj,ηi)=Ki,$(24)

subject to boundary conditions

$Ur+1(ξN,ηj)=Ur+1(ξ0,ηj)=Ur+1(ξi,ηM)=Ur+1(ξi,η0)=0.$(25)

Substituting Eqn. (25) into Eq. (24) we get

$∑p=1N−1D^jp2Ur+1(ξp,ηi)+∑q=1M−1d^iq2Ur+1(ξj,ηq)+λ4eUr(ξj,ηi)Ur+1(ξj,ηi)=Ki,$(26)

$⇒AUr+1=Ki$(27)

Eq. (27) can be written in a matrix form as

$A0,0A0,1⋯A0,MA1,0A1,2⋯A1,M⋮⋮⋱⋮AN−1,0AN−1,1⋯AN−1,MAN,0AN,1⋯AN,MUr+1,0Ur+1,1⋮Ur+1,N−1Ur+1,N=K0K1⋮KM−1KM$(28)

where

$Aii=D^2+d^iiI+diag[λ4eUr,i],Aij=d^ij2I,Ki=λ4eUr,i∘[Ur,i−i].$

The column vector i = [ 1 1 ⋯ 1]T and the Hamadard product AB is the element-wise multiplication of matrices A and B of the same order. Boundary conditions are applied to the system (28) as follows:

$I0⋯000A^1,1⋯A^1,M−10⋮⋮⋱⋮⋮0A^N−1,1⋯A^N−1,M−1000⋯0IU^r+1,0U^r+1,1⋮U^r+1,N−1U^r+1,N=0K^1⋮K^M−10$(29)

where I and 0 are identity and zero matrices, respectively, of order N × M and

$A^ij=10⋯00Aij00⋯01andK^i=0Ki0$(30)

## 2.3 Chebyshev spectral collocation method

As mentioned before, the exact solution of the two dimensional Bratu problem is unknown. As a basis of comparison, we will use the Chebyshev spectral collocation method, which uses Kronecker multiplication, abbreviated CSCM-K in this work, to solve the transformed Eq. (21) subject to boundary conditions Eq. (22). We intend to use the CSCM-K because of its known high order accuracy [31]. In multi-dimensional problems, the spectral collocation methods make use of Kronecker products to discretize the differential operators. It is worth noting that though the BSQLM uses spectral collocation, its main difference with the CSCM-K is the manner in which the two methods treat non-linearity.

#### Definition 2.1

If A and B are of dimensions p × q and r × s, respectively, then the Kronecker product AB is the matrix of dimension pr × qs with p × q block form where the i,j block is ai,jB, that is

$A⊗B=a1,1Ba1,2B⋯a1,q−1Ba1,qBa2,1Ba2,2B⋯a2,q−1Ba1,qB⋮⋮⋱⋮⋮ap−1,1Bap−1,2B⋯ap−1,q−1Bap−1,qBap,1Bap,2B⋯ap,q−1Bap,qB$(31)

The matrix AB is not dense, at the same time not as sparse as matrices from traditional methods, like the finite element or finite difference methods. Since Eq. (22) is highly non-linear, its solution is computed iteratively using the linear system

$Mus+1=fs,s=0,1,2,⋯$(32)

where M = I2 + 2I, fs = –exp(us) and us is the current iteration. I2 and 2I denote second order spectral differentiation in the x and y directions respectively. AB can be easily computed using the MATLAB command kron(A, B). The homogeneous boundary conditions Eq. (22) are implemented by deleting the first and the last rows and columns of the spectral differentiation matrix.

## 3 Results and discussion

Solving Eq. (7) for different values of the parameter λ (shown in Table 1), we obtain two values of ω labeled ω1 and ω2. Substituting the values of ω1 and ω2 into Eq. (6) and take the maximum of u(x,y)(denoted umax1 and umax2) we get results, which are shown in Table 1. We consider the values of λ as done in [19].

Table 1

The maximum values of the solution u(x, y) by the WRM, CSCM-K and BSQLM for different values of λ

A graphical representation of the results in Table 1 is done in Figure 1

Figure 1

The maximum values of u(x,y) versus λ.

Figure 1 shows that the analytical solution has two branches (lower and upper). It is also clear that the Bratu problem has two solutions for 0 < λ < λc, one solution for λ = λc and no solution for λ > λc. The mesh plots of the lower branch (λ = 5, ω1 = 2.811554938) and upper branch (λ = 5, ω2 = 7.548098106) are shown in Figure 2.

Figure 2

Mesh plots of the upper and lower solutions for selected values of λ and ω

In Table 2, we compare the values of umax of the three methods, WRM, CSCM-K and BSQLM. We consider the values of the parameter λ as done in [19]. We also consider the solution in the domain subdivided into equal 4 × 4 sub-regions. Since the exact solution of the two dimensional Bratu problem is not known in literature, we use the results from WRM and CSCM-K as our basis of comparison to the BSQLM results. The results in Table 2 show that the BSQLM solution agree to 5 decimal places with the results from CSCM-K, which is known to be highly accurate method.

Table 2

The maximum values of the solution u(x, y) by the WRM, CSCM-K and BSQLM for different values of λ

Rounding off the results in Table 2 to the nearest 5 decimal places for λ (= 1, 2, 3, 4, 5, 6) we get the results in Table 3. These results compare very well with the results obtained by [22] after solving the two dimensional Bratu problem using the optimal homotopy analysis (oHAM) and wavelet homotopy analysis (wHAM) and the iterative wHAM.

Table 3

The maximum values of u(x, y) for various values of λ

We solve the two dimensional Bratu problem using BSQLM for λ = 1 and 10 × 10 sub-regions of the problem domain [0, 1] × [0, 1]. A graphical representation of the solution is a mesh plot in Figure 3 which agrees with the solution plot in [22]. With the fuel ignition model as one of the physical applications of the Bratu problem, taking u(x,y) to represent temperature, the results in Figure 3 show that there is a continuous decrease of temperature from the midpoint towards the boundary. Table 4 compares umax from BSQLM, CSCM-K and FDM for different values of the λc as done in [19]. For all the cases, we consider equal sub-regions and a tolerance level of 10–6. The results show that the BSQLM produces results in good agreement with CSCM-K. Due to lack of an exact solution, it is impossible to directly compute the accuracy of the BSQLM in solving the two dimensional Bratu problem. We now compare the two methods, BSQLM and CSCM-K in terms of speed of convergence and computational efficiency.

Figure 3

Solution of the Bratu problem using BSQLM for λ = 1.

Table 4

The maximum values of u(x, y) for various values of λc and N × N.

Table 5 shows a comparison between BSQLM and CSCM-K in terms of runtime in seconds and number of iterations for the solutions to converge within a tolerance level of 10–15 for different values of equal sub-regions N × N. In all the cases, we consider the constant parameter λ = 1. It is worth noting that after the 10 × 10 sub-region, the BSQLM solution takes less time to converge than the CSCM-K solution. This shows that the BSQLM is more computationally efficient than CSCM-K. Moreover, as the subdivisions are made finer, the BSQLM needs less iterations to converge than the CSCM-K, which needs a constant 12 iterations to converge. This proves that the BSQLM is faster than CSCM-K.

Table 5

Runtime and the number of iterations

## 4 Conclusion

We solved the two dimensional Bratu problem using the bivariate spectral collocation method (BSQLM) and the Chebyshev spectral collocation method that uses Kronecker multiplication (CSCM-K). Both methods produced solutions which converge to the lower solution. We compared the results with those from finite difference method (FDM) and weighted residual method (WRM) in literature. We observed that results from BSQLM and CSCM-K are in close agreement with the results in Table 2. Rounding-off the results in Table 2 to 5 decimal places for selected values of λ, we get results in Table 3, which are in excellent agreement with the results obtained using the oHAM, wHAM and iterative HAM. This proves that the BSQLM is capable of producing reliable results as it compares well with the wHAM, which is known to have a high computational efficiency. From the results in Table 5, we conclude that BSQLM is faster and more computationally efficient than CSCM-K.

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Accepted: 2018-05-30

Published Online: 2018-08-20

Conflict of InterestConflict of Interests: The authors declare that there is no conflict of interests regarding the publication of this paper.

Citation Information: Open Physics, Volume 16, Issue 1, Pages 554–562, ISSN (Online) 2391-5471,

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