Application of HPM for determination of an unknown function in a semi-linear parabolic equation Malihe Rostamian 1 and Alimardan Shahrezaee 1 1,2

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1 ISSN 76659, England, UK Journal of Information and Computing Science Vol., No., 5, pp. - Application of HPM for determination of an unknon function in a semi-linear parabolic equation Malihe Rostamian and Alimardan Shahrezaee, Department of Mathematics, Alzahra university, Vanak, Post Code 98, Tehran, Iran. (Received November,, accepted February, 5) Abstract. In this ork, the homotopy perturbation method, a poerful technique, is applied to obtain an unknon time-dependent function in a semi-linear parabolic equation ith given initial and boundary conditions. This kind of problem plays a very important role in many branches of science and engineering. Using the homotopy perturbation method a rapid convergent sequence can be constructed hich tends to the exact solution of the problem. Some examples are presented to illustrate the strength of the method. Keyords: Inverse problem, Semi-linear parabolic equation, Homotopy perturbation method.. Introduction Most of physical and engineering problems are nonlinear and in most cases it is difficult to solve them, especially analytically. A number of analytical methods are available in the literature for the investigation of these problems, such as Adomian decomposition method [, ], the -expansion method [], the homotopy analysis method [], the variational iteration method [7-] and the homotopy perturbation method (HPM) [-]. The HPM as proposed by Ji-Huan He in 999. The essential idea of this method is to introduce a homotopy parameter, say m, hich takes values from to. When m, the systems of equations usually reduced to a sufficiently simplified form, hich normally admits a rather simple solution. As m is gradually increased to, the system goes through a sequence of deformations, the solution for each of hich is close to that at the previous stage of deformation. Eventually at m, the systems takes the original form of the equation and the final stage of deformation gives the desired solution. One of the most considerable features of the HPM is that usually just fe perturbation terms are sufficient for obtaining a reasonably accurate solution. The HPM is an effective solution method for a broad class of problems. This technique as applied to nonlinear oscillators ith discontinuities [], nonlinear ave equations [5], nonlinear boundary value problems [6], a nonlinear convection- radiative cooling equation, a nonlinear heat equation [7], limit cycle and bifurcation of nonlinear problems [8, 9], nonlinear fractional partial differential equations [], inverse heat conduction problem [] and some other subjects [-6]. In this ork, e consider the inverse problem of finding a pair of function ( T, p ) in the folloing semilinear parabolic equation: T X, t T X, t p ( t ) T ( X, t ) ( X, t ); X, t t max, () t ith initial and boundary conditions: T ( X,) f ( X ); X, () T ( X, t ) h( X, t ); X, t t max, () and an additional condition as an over specification at a point in the spatial domain in the folloing form: T ( X, t ) E ( t ); X, t t, () here is Laplace operator, t max is final time, [,] d is spatial domain of the problem for d,,, X ( x,, x d ), is the boundary of and, f, h and E are knon functions. The existence, uniqueness and continuous dependence of the solution upon the data for this problem are demonstrated in [7-]. These kinds of problems have many important applications in heat transfer, thermoelasticity, control theory and chemical diffusion. Equation () can be used to describe a heat transfer process ith a source parameter pt () and () to represent the temperature T ( X, t ) at a specific point X in the spatial domain at max Corresponding author. address: ashahrezaee@alzahra.ac.ir. Published by World Academic Press, World Academic Union

2 Journal of Information and Computing Science, Vol. (5) No., pp - 5 any time. In [, ] the finite difference techniques are used to approximate the solution of this problem. The Adomian decomposition method for the problem ()-() is proposed in []. Authors of [5, 6] applied the variational iteration method to obtain the analytical solution for this problem. Also Sinc-collocation method has been used in [7] for solving the one dimensional parabolic inverse problem ith a source control parameter. In this paper, e use the HPM to derive an analytical solution for the problem ()-().The organization of the paper is as follos: In Section, the homotopy perturbation method is presented. Section is devoted to some examples. Conclusion is finally discussed in Section.. Homotopy perturbation method (HPM) To clarify the basic ideas of HPM, consider the folloing nonlinear differential equation: A( T ) f ( r) ; r, (5) subject to the boundary condition: T B ( T, ) ; r, (6) n here T T ( X, t ) is the dependent variable to be solved, A is a general differential operator, B is a boundary operator and f ( r ) is a knon analytic function. The operator A can be divided into to parts, hich are L and N, here L is a linear and N is a nonlinear operator. Equation (5) can be, therefore, ritten as: L( T ) N ( T ) f ( r). (7) By using homotopy technique, one can construct a homotopy v ( r, m): [,] (8) hich satisfies H ( v, m) ( m)[ L( v ) L( T)] m[ A( v ) f ( r)], (9) or H ( v, m) L( v ) L( T ) ml( T ) m[ N ( v ) f ( r)], () here, m is an embedding parameter and T is the initial approximation of equation (5) hich satisfies the boundary conditions. Clearly, from equations (9) and (), e have H ( v,) L( v ) L( T), () H ( v,) A( v ) f ( r). () Changing the process of m from zero to unity is just that of v ( r, m ) changing from T ( r ) to T ( r ). In topology, this is called deformation and also, L( v ) L( T) are called homotopic. According to the homotopy perturbation method, the parameter m is considered as a small parameter and the solution of equations () and () can be given as a series in m in the form [-]: v T mt m T, () and setting m results in the approximate solution of equation (5) as: T lim v T T T. () m If e limit the sum to the first n components, e obtain so-called n order approximate solution of equation (): T T T T n. (5) The major advantage of HPM is that the perturbation equation can be freely constructed in many ays (therefore is problem dependent) by homotopy in topology and the initial approximation can also freely selected. For the convergence of the series obtained via HPM, e recall Banach's theorem: Theorem. Assume that X is a Banach space and N : X X is a nonlinear mapping and suppose that v, v X ; N ( v ) N ( v ) v v,. Then N has a unique fixed point. Furthermore, the sequence V n N ( V n) JIC for subscription: publishing@wau.org.uk

3 6 Malihe Rostamian et.al. : Application of HPM for determination of an unknon function in a semi-linear parabolic equation ith an arbitrary choice of V X converges to the fixed point of N and k V V V V. k l j jl The sequence generated by HPM ill be regarded as: n V T, V N ( V ), N ( V ) T, i,,. n n n i i According to the above theorem, for the nonlinear mapping N a sufficient condition for the convergence of HPM is strictly contraction of N. Before applying the HPM to equation (), e employ to folloing transformations: t ( X, t ) T ( X, t )exp( p( s ) ds ), (6) t r( t ) exp( p( s ) ds ). (7) Transformation (6) allos us to eliminate the unknon term pt () from equation () and to obtain a ne non-classic partial differential equation hich has suitable form to apply the HPM. No using trensformations (6) and (7), e can rite ()-() as follos: X, t X, t r ( t ) ( X, t ); X, t t max, (8) t ( X,) f ( X ); X, (9) ( X, t ) r( t ) h( X, t ); X, t t, () ( X, t ) r( t ) E ( t ); X, t t. () Assume Et ( ), then the later is equivalent to: ( X, t ) rt ( ). Et () () According to HPM, e construct the folloing homotopy [-] (, ) (, ) (, ) { (, ) (, ) X t m X t X t (, )}. t t E () t t () The solution of equation () is assumed in the form [-]: m m m. () Substituting () into () and collecting coefficients of the same poer of m yield: m :, t t (5) m : ( X, t ), t E t (6) m : ( X, t ), t E (7) (8) We let (, ) ( ), (9) then all the above linear equations can be easily solved. The solution of () can be obtained by putting m in equation () as follos:. () No from (6), e compute: ( X, t ) T ( X, t ), rt () () and from (7), e obtain: r() t pt ( ), rt () () max max JIC for contribution: editor@jic.org.uk

4 Journal of Information and Computing Science, Vol. (5) No., pp - 7 here rt () is given by (). The numerical results in Section indicate that the proposed scheme is efficient.. Numerical examples In this section, e present some examples to sho the high accuracy of HPM for solving the inverse problem ()-(). Example. Consider the folloing problem [5, 9,,, ]: T T p( t ) T ( ( t ) )exp( t )(cos( x ) sin( x )); x, t, () t T ( x,) cos( x ) sin( x ); x, T t t t (, ) exp( );, T t t t (, ) exp( );, T (.5, t ) exp( t ); t. The exact solution of this problem is: T ( x, t ) exp( t )(cos( x ) sin( x )), P( t ) t. Using the equation (6)-(9), e find ( x, t ) f ( x ) cos( x ) sin( x ), t ( x, t ) ( t t )(cos( x ) sin( x )), t ( t t) ( x, t ) (cos( x ) sin( x )), Then from (), e have the approximate solution in a series form as: t ( t t) t ( x, t ) ( ( t t ) )(cos( x ) sin( x )) t exp( t t )(cos( x ) sin( x )). No from () and () the exact values of T ( x, t ) and pt () can be obtained. This is the same as obtained by Adomian's decomposition method and the variational iteration method [5]. We can compute n order approximate solution of equation () from (5) as: t t n ( t t ) ( t t ) t ( x, t ) ( ( t t ) )(cos( x ) sin( x )). n! Then from () and () the n order approximate values of T ( x, t ) and pt () can be obtained. Tables and sho the comparison of absolute error of several methods in approximating T ( x,) and pt (), respectively, for problem. As e see the HPM has good accuracy in comparison ith the other methods of [,, ]. In HPM e take n 5. Figures and presents the exact and numerical values of T ( x,) and pt (),respectively. In HPM e take n and n 5. JIC for subscription: publishing@wau.org.uk

5 8 Malihe Rostamian et.al. : Application of HPM for determination of an unknon function in a semi-linear parabolic equation Table. Comparison of absolute error of the several techniques in approximating T ( x,) for test problem. x HPM CFDM[] SaulyevII [] MOL[] Table. Comparison of absolute error of the several techniques in approximating pt () for test problem. t HPM CFDM[] SaulyevII [] MOL[] Example. In this example, let us consider the folloing problem [, 5]: T 5 T p( t ) T ( 5 t )exp( t )sin( ( x y )); t 6 x, y, t, T ( x, y,) sin( ( x y )); x, y, T (, y, t ) exp( t )sin( y ); y, t, T (, y, t ) exp( t )sin( ( y )); y, t, T ( x,, t ) exp( t )sin( x ); x, t, T ( x,, t ) exp( t )sin( ( x )); x, t, T (.,., t) exp( t)sin(. ); t. () JIC for contribution: editor@jic.org.uk

6 T(x,) p(t) Journal of Information and Computing Science, Vol. (5) No., pp Exact HPM,n= HPM,n=5. Exact HPM,n= HPM,n= x t Fig. : The exact and numerical values of T ( x,). Fig. : The exact and numerical values of pt ( ). The exact solution of this problem is: T ( x, y, t ) exp( t )sin( ( x y )), P ( t ) 5 t. Using the equation (6)-(9), e compute ( x, y, t ) f ( x, y ) sin( ( x y )), 5 ( x, y, t ) t sin( ( x y )), 5 ( t ) ( x, y, t ) sin( ( x y )), So from (), e have the approximate solution in a series form as: 5 ( t ) 5 5 ( x, y, t ) ( ( t ) )sin( ( x y )) exp( t )sin( ( x y )). One can compute the exact values of T ( x, y, t ) and pt () from () and (). This result is the same as obtained by Adomian's decomposition method [] and the variational iteration method [5]. The n order approximate solution of equation () can be obtained as follos: 5 5 n ( t ) ( t ) 5 (, ) ( ( ) x t t )sin( ( x y )). n No from () and () the n order approximate values of T ( x, y, t ) and pt () can be obtained. Tables and sho the comparison of absolute error of several methods in approximating T ( x, y,) and pt (), respectively, for problem. As e see the HPM has good accuracy in comparison ith the other methods of []. In HPM e take n. Figures, and 5 presents the exact and numerical values of T ( x, y,) and pt (). In figure, e put n and in figure e take n. JIC for subscription: publishing@wau.org.uk

7 T(x,y,) T(x,y,) Malihe Rostamian et.al. : Application of HPM for determination of an unknon function in a semi-linear parabolic equation Table. Comparison of absolute error of the several techniques in approximating T ( x, y,) for test problem. x HPM (,) F.E. [] (9,9) F. I. [] (,9) ADI[] Table. Comparison of absolute error of the several techniques in approximating pt () for test problem. t HPM (,) F.E. [] (9,9) F. I. [] (,9) ADI[] Exact HPM, n= y x y x Example. In this example, consider [6]: Fig. : The exact and numerical values of T ( x, y,). JIC for contribution: editor@jic.org.uk

8 T(x,y,) T(x,y,) p(t) Journal of Information and Computing Science, Vol. (5) No., pp - T t T p t T yz x y y t y y t t x x y z t ( ) ( 5 6)exp( );,,,, T x y z zy x x y z (,,,) exp( );,,, T y z t zy t y z t (,,, ) exp( );,,, T y z t zy t y z t (,,, ) exp( );,,, T ( x,, z, t ) ; x, z, t, T x z t z x t x z t (,,, ) exp( );,,, T ( x, y,, t ) ; x, y, t, T x y t y x t x y t (,,, ) exp( );,,, T (,,, t ) 8exp( t ); t Exact HPM,n= HPM,n= 5 Exact HPM, n= y.6.. x.. y.6.. x t Fig. : The exact and numerical values of T ( x, y,). Fig. 5: The exact and numerical values of pt ( ). The exact solution of this problem is: T ( x, y, z, t ) zy exp( x t ), P t t t According to the equations (6)-(9), e obtain: ( x, y, z, t ) zy exp( x ), ( ) 5. 5 t ( x, y, z, t ) zy ( t t )exp( x ), 5 t ( t t ) x y z t zy x (,,, ) ( )exp( ), Thus from (), e have in series from: 5 t ( t t ) 5 t 5 t ( x, y, z, t ) ( ( t t ) ) zy exp( x ) zy exp( x t t ). No from equations () and () the exact values of T ( x, y, z, t ) and pt () can be obtained. This problem bas been solved by the variational iteration method in [6].. Conclusion JIC for subscription: publishing@wau.org.uk

9 Malihe Rostamian et.al. : Application of HPM for determination of an unknon function in a semi-linear parabolic equation In this paper, the HPM has been successfully employed to obtain an unknon parameter in a semi-linear partial differential equation ith given initial and boundary conditions. This method constructs the solution of the problem as a rapid convergent series solution. Implementation of this method is easy and calculation of successive approximations is straightforard. Comparing ith some numerical methods [,,,], the HPM solves the problem ithout any discretization of the variables, therefore is free from rounding off errors in the computational process. Also, it does not require large computer memory or time. The HPM provide the solution in a closed form hile the mesh point techniques [,,, ] provide the approximation at mesh points only. Using Adomian decomposition method and the variational iteration method, the same results ill be obtained. The advantage of the proposed method over Adomian decomposition method is that homotopy perturbation technique obtain the solution of the problem ithout calculating of Adomian's polynomials. Also computing the successive terms in HPM is much easier than the variational iteration method. The results sho that the HPM is a poerful mathematical tool for finding the analytical solution of inverse problem. 5. References [] G. Adomian, A revie of the decomposition method in applied mathematics, Journal of Mathematical Analysis and Applications 5 (988), pp. 5. [] R. Grzymkoski, M. Pleszczynski and D. Slota, Comparing the Adomian decomposition method and Rung- Kutta method for the solutions of the stefan problem, Journal of Computer Mathematics 8 (6), pp. 9. [] A.V. Karmishin, A.I. Zhukov and V.G. Kolosov, Methods of Dynamics Calculation and Testing for Thin-Walled Structures, Mashinostroyenie, Mosco, 99. [] S.J. Liao, Beyond Perturbation: Introduction to the Homotopy Analysis Method, Chapman \& Hall/CRC Press, Boca Raton,. [5] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph. D. thesis, Shanghai Jiao Tong University, Shanghai, 99. [6] S.J. Liao, Notes on the homotopy analysis method: some definitions and theorems, Communications in Nonlinear Science and Numerical Simulation (9), pp [7] J.H. He, Variational iteration method-a kind of nonlinear analytical technique: some examples, International Journal of Nonlinear Mechanics (999), pp [8] J.H. He, Variational iteration method for autonomous ordinary differential system, Applied Mathematics and Computation (), pp. 5-. [9] J.H. He and X.H. Wu., Construction of solitary solution and compaction-like solution by variational iteration method, Chaos, Solitons and Fractals 9 (6), pp. 8-. [] J.H. He, Variational approach for nonlinear oscillators, Chaos, Solitons and Fractals (7), pp. -9. [] J.H. He, Homotopy perturbation technique, Comput. Methods. Appl. Mech. Engrg. 78 (999), pp [] J.H. He, A coupling method of a homotopy technique and a perturbation technique for non-linear problems, Internat. J. Non-linear Mach. 5 (), pp. 7-. [] J.H. He, Homotopy perturbation method: A ne nonlinear analytical technique, Appl. Math. Comput. 5 (), pp. 79. [] J.H. He, The homotopy perturbation method for nonlinear oscillators ith discontinuities, Appl. Math. Comput. 5 (), pp. 87. [5] J.H. He, Application of homotopy perturbation method to nonlinear ave equations, Chaos, Solitons and Fractals 6 (5), pp [6] J.H. He, Homotopy perturbation method for solving boundary value problems, Physics Leters A 5 (6), pp [7] M. Rafei, D.D. Gangi and H. Daniali, Solution of the epidemic model by homotopy perturbation method, Applied Mathematics and Computation 87 (7), pp [8] J.H. He, Limit cycle and bifurcation of nonlinear problems, Chaos, Solitons and Fractals 6 (5), pp [9] J.H. He, Periodic solutions and bifurcations of delay-differential equations, Phys. Lett. A 7 (5), pp. 8-. [] Q. Wang, Homotopy perturbation method for fractional Kdv-Burgers equation, Chaos, Solitons and Fractals 5 (8), pp [] Edyta Hetmaniok, Iona Noak, Damian Slota and Roman Witula, Application of the homotopy perturbation method for the solution of inverse heat conduction problem, International Communications in Heat and Mass Transfer 9 (), pp. -5. JIC for contribution: editor@jic.org.uk

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