MPEMathematical Problems in Engineering1563-51471024-123XHindawi Publishing Corporation92608610.1155/2009/926086926086Research ArticleAdomian Decomposition Method for a Nonlinear Heat Equation with Temperature Dependent Thermal PropertiesBokhariAshfaque H.1MohammadGhulam2MustafaM. T.1ZamanF. D.1RagabSaad A.1Department of Mathematics and StatisticsKing Fahd University of Petroleum and MineralsDhahran 31261Saudi Arabiakfupm.edu.sa2Department of MathematicsNational College of Business Administration and EconomicsLahorePakistanncbae.edu.pk20090609200920092612200820042009140720092009Copyright © 2009This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

The solutions of nonlinear heat equation with temperature dependent diffusivity are investigated using the modified Adomian decomposition method. Analysis of the method and examples are given to show that the Adomian series solution gives an excellent approximation to the exact solution. This accuracy can be increased by increasing the number of terms in the series expansion. The Adomian solutions are presented in some situations of interest.

1. Introduction

In the classical model of the heat equation, the thermal diffusivity and thermal conductivity of the medium are assumed to be constant. In some media such as gases, these parameters are proportional to the temperature of the medium giving rise to a nonlinear heat equation of the following form :

C(x)ut=λx(kuux), where C is the conductivity, k is diffusivity, and λ is a constant.

However, in some situations the diffusivity is proportional to uα, which gives rise to a more general nonlinear heat equation

C(x)ut=λx(uαux). In this paper we investigate the nonlinear heat equation

ut=x(f(u)ux), with f(u)=um, using the Adomian decomposition method. This method was presented by Adomian to solve algebraic, differential, integrodifferential equations and stochastic problems . In these papers Adomian presented the so-called decomposition method in which the problem is split into linear (solvable) and nonlinear part. By assuming that the solution admits a power series representation, the nonlinear contribution to the solution is obtained in the form of “Adomian polynomials” . Alternative methods of calculating Adomian polynomials have been discussed by Babolian and Javadi  and Wazwaz . For the convergence of the Adomian method, see . For a detailed treatment and applications of the Adomian decomposition method one may refer to . Chiu and Chen  have applied the Adomian method to study fin problem with variable conductivity. Wazwaz in  established an algorithm for calculating Adomian polynomials that depend mainly on algebraic and trigonometric identities and on Taylor’s expansion. A feature of this method is that it involves less formulas and is straightforward to implement. The reader is referred to [10, Section  2] for details of algorithm and its connection with earlier approach of Adomian . We will use the modified Adomian algorithm given by Wazwaz  to find the Adomian solutions to our models of nonlinear heat equation with temperature dependent diffusivity.

2. Method of Solution

Introducing the operator Lt=/t, (1.3) takes the form

Ltu(x,t)=[f(u)ux2+f(u)uxx]. We solve (2.1) subject to the initial condition

u(x,0)=g(x). Applying inverse operator Lt-1  to both sides of (2.1) yields

u(x,t)=u(x,0)+Lt-1[(f(u)ux2+f(u)uxx)]. The desired series solution by Adomian decomposition method is given by (cf.  for details)

u(x,t)=n=0un(x,t), and u1,u2,u3, are calculated from recursive relation

u0=u(x,0),un+1=Lt-1[(An)],      n0, where  An are the Adomian polynomials for the nonlinear operator

F(u(x,t))=f(u)ux2+f(u)uxx. The formulas that can be used to generate Adomian polynomials are discussed by Adomian in . Here we employ the algorithm of Wazwaz  to calculate Adomian polynomials, which seems quite natural and suited for implementation by software.

3. Applications and Results

We consider the nonlinear heat equation

ut=x(f(u)ux),u(x,0)=g(x) with power nonlinearity f(u)=um. We are interested in investigating the case of power nonlinearity due to the fact that this assumption is made in most of the applied nonlinear problems of heat transfer and flows in porous media. For instance, f(u)=u-1/2 corresponds to fast diffusion processes of plasma diffusion and thermal expulsion of liquid Helium . The diffusivity f(u)=u2 is used to model process of melting and evaporation of metals . For the initial temperature profile, we consider typical cases like g(x) a quadratic function or g(x)=e-ax2 or g(x)=sech2x which corresponds to soliton like initial profile.

Case A (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M26"><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mi>a</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mo>+</mml:mo><mml:mi>b</mml:mi><mml:mi>x</mml:mi><mml:mo>+</mml:mo><mml:mi>c</mml:mi></mml:math></inline-formula>).

The Adomian solution u(x,t) for general a, b, c, and  m can be obtained from authors as Mathematica file. Some particular cases for a, b, c, and  m  are considered as follows.

(i)  a=b=c=1  and  m=2.

The Mathematica code to obtain Adomina solution in this case consists of the following commands:

f[n_]=i=0n-1ui[x]αi+O[α]n,f1d[n_]=i=0n-1xui[x]αi+O[α]n,f2d[n_]=i=0n-1x,xui[x]αi+O[α]n, maximum number of polynomials and solution terms: k=5 Finding Adomian polynomials: apoly=mf[k]m-1f1d[k]2//Simplifybpolynomial=f[k]mf2d[k]. Making vector of admian polynomials: v=CoefficientList[coeffpoly,α]. Finding solution u(x,t): u0[x_]=a*x2+b*x+cDo[ui[x_]=0tv[[i]]dt,{i,1,k}]u[x_,t_]=u0[x]Do[u[x_,t_]=u[x,t]+ui[x],{i,1,k}]u[x,t]a=1;b=1;c=1m=2u[x,t].   The Adomian solution obtained is u(x,t)=1+x+x2+2t2(1+x(1+x))×(24+45x+185x2+4x(25+70x2)+4(2+25x2+35x4))+13t3(1+x(1+x))×(60+1860x+8x(1860+7995x2)+2(570+8370x2)+8x(1665+10770x2+12825x4)+4(720+14490x2+29370x4)+8(75+1665x2+5385x4+4275x6))+13t4(1+x(1+x))×(8160+11310x+180390x2+4x(50610+318390x2)+4(13380+414000x2+1172820x4)+4x(160950+1507380x2+2417550x4)+16x(20370+222060x2+585450x4+429000x6)+4(17970+605070x2+2705190x4+2807850x6)+16(600+20370x2+111030x4+195150x6+107250x8))+160t5(1+x(1+x))×(15120+1275120x+2(486360+16082040x2)+8x(5172000+45672540x2)+4(3226560+145378920x2+558401640x4)+8x(36485460+477387720x2+1007253300x4)+64x(8126460+123096840x2+424022700x4+390053400x6)+8(5708700+279944820x2+1685856660x4+2244119100x6)+32x(5402520+86621760x2+373392000x4+591280800x6+309309000x8)+16(2417760+119127600x2+868513680x4+1945018800x6+1317980400x8)+32(113400+5402520x2+43310880x4+124464000x6+147820200x8+61861800x10))+t(1+x(1+x))(2+2(1+5x(1+x))). The solutions in Figure 1 increase algebraically as is expected from algebraic behavior of initial condition and the form of   f(u).

(ii) a=b=c=1  and  m=-2 (Figure 2).

As the diffusivity in this case is decreasing function of   u, the solution exhibits the change in the quadratically increasing initial temperature.

(iii) a=b=c=1  and  m=1/2 (Figure 3).

(a) Graph of Adomian solution for {x,0,5},{t,0,1}. (b) Graph Adomian solution for {x,-5,5},{t,0,1}.

Graph of Adomian solution for the range {x,-2,1},{t,0,1}.

Graph of Adomian solution for the range {x,-2,1},{t,0,1}.

Case B (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M45"><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>e</mml:mi></mml:mrow><mml:mrow><mml:mo>-</mml:mo><mml:mi>a</mml:mi><mml:msup><mml:mrow><mml:mi>x</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:msup></mml:math></inline-formula>).

The Adomian solution for general a,m can be obtained from authors as Mathematica file. Some particular cases are considered as follows.

(i)   a=2 and m=2

u(x,t)=e-2x2+4e-6x2t(-1+12x2)+8e-10x2t2(11-400x2+1200x4)+323e-14x2t3(-315+22692x2-181552x4+291648x6)+323e-18x2t4(16425-1947360x2+28962720x4-115402752x6+123607296x8)+12815e-22x2t5(-1326840+233242200x2-5491343520x4+38961513344x6-99063148800x8+78562446336x10+945(-1+20x2)). Figure 4 displays how the bell-shaped initial temperature interacts with quadratic dependence of diffusivity.

(ii) a=2 and m=-2 (Figure 5).

u(x,t)=e-2x2+4e2x2t(-1-4x2)+8e6x2t2(-5-96x2-144x4)+323e10x2t3(-91-4028x2-19120x4-17600x6)+323e14x2t4(-3287-260480x2-2523104x4-6375936x6-4202240x8)+12815e18x2t5(-191704-23954712x2-390296736x4-1877037696x6-3158528256x8-1613177856x10+945(-1+20x2)).

(iii) a=2 and m=1/2 (Figure 6).

u(x,t)=e-2x2+4(e-2x2)3/2t(-1+6x2)+8e-4x2t2(5-76x2+96x4)+323(e-2x2)5/2t3(-2194+3009x22-4615x4+2850x6)+323e-6x2t4(20312-43365x2+233406x4-337716x6+131760x8)+12815(e-2x2)7/2t5(-43338916+13476579x28-27882147x42+34527269x6-30761241x8+8579214x10+945(-1+20x2)13476579x28).

(a) Graph of solution for the range {x,-5,5},{t,0,1}. (b) Graph of solution for the range {x,-2,2},{t,0,1}. (c) Graph for fixed t=0.5 for the range {x,-1,1}.

(a) Graph for the range {x,-.01,.01},{t,0,1}. (b) Graph for the range {x,-1.5,.01},{t,0,1}.

(a) Graph for the range {x,-1,1},{t,0,1}. (b) Graph for the range {x,-10,10},{t,0,1}.

Case C (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M64"><mml:mi>g</mml:mi><mml:mo stretchy="false">(</mml:mo><mml:mi>x</mml:mi><mml:mo stretchy="false">)</mml:mo><mml:mo>=</mml:mo><mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mtext>sech</mml:mtext></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup></mml:mrow><mml:mo> </mml:mo><mml:mrow /></mml:mrow><mml:mi>x</mml:mi></mml:math></inline-formula>).

The Adomian solution for general   m can be obtained from authors as Mathematica file. Some particular cases are considered as follows.

(i)   m=2.

u(x,t)=sech(x)2+2t(-4+3cosh(2x))sech(x)8+3t2(161-178cosh(2x)+25cosh(4x))sech(x)14+t3(-54900+71641cosh(2x)-18772cosh(4x)+1519cosh(6x))×sech(x)20+14t4(35318621-50550350cosh(2x)+18047504cosh(4x)-2916178cosh(6x)+160947cosh(8x))×sech(x)26+120t5(-35893153056+54495231330cosh(2x)-23506173696cosh(4x)+5488700877cosh(6x)-621401568cosh(8x)+25573713cosh(10x))sech(x)32. Here the initial condition is soliton like. This is reflected in the Figure 7 as the diffusivity varies quadratically.

(ii) m=-2 (Figure 8).

u(x,t)=-2tcosh(2x)-t2cosh(x)2(1-2cosh(2x)+9cosh(4x))-13t3cosh(x)4(-44+85cosh(2x)-76cosh(4x)+275cosh(6x))-112t4cosh(x)6(2865-5862cosh(2x)+5968cosh(4x)-5178cosh(6x)+16415cosh(8x))-160t5cosh(x)8×(-303864+606738cosh(2x)-616768cosh(4x)+638373cosh(6x)-544328cosh(8x)+1575369cosh(10x))+sech(x)2.

(iii)  m=1/2 (Figure 9).

u(x,t)=sech(x)2+32t2(56-52cosh(2x)+4cosh(4x))sech(x)8+18t4×(58894158-37503834cosh(2x)+232028cosh(4x)-764174cosh(6x)+27458cosh(8x))×sech(x)14+2t(-52+32cosh(2x))sech(x)4sech(x)2+12t3(-379178+8600516cosh(2x)-71638cosh(4x)+47516cosh(6x))×sech(x)8(sech(x)2)3/2+140t5×(-22986251157128+31585649589128cosh(2x)-250111610132cosh(4x)+2695647273256cosh(6x)-64605399128cosh(8x)+1429869256cosh(10x))×sech(x)12(sech(x)2)5/2.

(a) Graph for the range {x,-1,1},{t,0,1}. (b) Graph for the range {x,-10,10},{t,0,1}.

(a) Graph for the range {x,-.01,.01},{t,0,1}. (b) Graph for the range {x,-1.5,.01},{t,0,1}. (c) Graph for fixed t=0.5 for the range {x,-0.5,0.01}.

(a) Graph for the range {x,-1,1},{t,0,1}. (b) Graph for the range {x,-10,10},{t,0,1}.

4. Conclusion

The Adomian decomposition method has been applied to obtain solutions of the heat equation with power nonlinearity in the diffusivity. The solutions are presented for some typical initial temperature profiles like a quadratic function or or e-ax2 or sech2x. The interaction of the initial temperature with diffusivity is also discussed for different cases of solutions investigated here.

Acknowledgment

The authors would like to thank King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, for the support and research facilities provided to complete this work.

Necati OziskM.Heat Conduction19932ndNew York, NY, USAJohn Wiley & SonsAdomianG.AdomianG. E.A global method for solution of complex systemsMathematical Modelling198454251263MR76851310.1016/0270-0255(84)90004-6ZBL0556.93005AdomianG.A new approach to nonlinear partial differential equationsJournal of Mathematical Analysis and Applications19841022420434MR75597310.1016/0022-247X(84)90182-3ZBL0554.60065AdomianG.RachR.Polynomial nonlinearities in differential equationsJournal of Mathematical Analysis and Applications198510919095MR79604410.1016/0022-247X(85)90178-7ZBL0606.34009AdomianG.A review of the decomposition method in applied mathematicsJournal of Mathematical Analysis and Applications19881352501544MR96722510.1016/0022-247X(88)90170-9ZBL0671.34053AdomianG.Solving Frontier Problems of Physics: The Decomposition Method199460Dordrecht, The NetherlandsKluwer Academic Publishersxiv+352Fundamental Theories of PhysicsMR1282283BabolianE.JavadiSh.New method for calculating Adomian polynomialsApplied Mathematics and Computation20041531253259MR206315910.1016/S0096-3003(03)00629-5ZBL1055.65068WazwazA.-M.wazwaz@sxu.eduA comparison between Adomian decomposition method and Taylor series method in the series solutionsApplied Mathematics and Computation19989713744EID2-s2.0-0000738773WazwazA.-M.A reliable modification of Adomian decomposition methodApplied Mathematics and Computation199910217786MR168285510.1016/S0096-3003(98)10024-3ZBL0928.65083WazwazA.-M.A new algorithm for calculating Adomian polynomials for nonlinear operatorsApplied Mathematics and Computation200011115369MR174590810.1016/S0096-3003(99)00063-6ZBL1023.65108WazwazA.-M.Approximate solutions to boundary value problems of higher order by the modified decomposition methodComputers & Mathematics with Applications2000406-7679691MR1776685ZBL0959.65090CherruaultY.Convergence of Adomian's methodKybernetes of Cybernetics and General Systems19891823138MR100997910.1108/eb005812ZBL0697.65051CherruaultY.AdomianG.Decomposition methods: a new proof of convergenceMathematical and Computer Modelling19931812103106MR126228610.1016/0895-7177(93)90233-OZBL0805.65057LesnicD.Convergence of Adomian's decomposition method: periodic temperaturesComputers & Mathematics with Applications2002441-21324MR1908267ZBL1125.65347ChiuC.-H.ChenC.-K.ckchen@mail.ncku.edu.twA decomposition method for solving the convective longitudinal fins with variable thermal conductivityInternational Journal of Heat and Mass Transfer2002451020672075EID2-s2.0-000258810210.1016/S0017-9310(01)00286-1DresnerL.Similarity Solutions of Nonlinear Partial Differential Equations198388Boston, Mass, USAPitmanviii+124Research Notes in MathematicsMR721168SaiedE. A.HusseinM. M.New classes of similarity solutions of the inhomogeneous nonlinear diffusion equationsJournal of Physics A1994271448674874MR129518610.1088/0305-4470/27/14/015ZBL0842.35002SaiedE. A.The non-classical solution of the inhomogeneous non-linear diffusion equationApplied Mathematics and Computation1999982-3103108MR165991210.1016/S0096-3003(97)10158-8WazwazA.-M.Exact solutions to nonlinear diffusion equations obtained by the decomposition methodApplied Mathematics and Computation20011231109122MR184671510.1016/S0096-3003(00)00064-3ZBL1027.35019