Modified Adomian Polynomial for Nonlinear Functional with Integer Exponent
Modified Adomian Polynomial for Nonlinear Functional with Integer Exponent
E. U. Agom , F. O. Ogunfiditimi (Department of Mathematics, University of Calabar, Calabar, Nigeria) (Department of Mathematics, University of Abuja, Abuja, Nigeria) Email: [email protected] Abstract :
Successful application of Adomian decomposition method (ADM) in solving problems in nonlinear ordinary and partial differential equations depend strictly on the Adomian polynomial. In this paper, we present a simple modified known Adomian polynomial for nonlinear polynomial functionals with index as integers. The simple modified Adomian polynomial was tested for nonlinear functional with index 3 and 4 respectively. The result shows remarkable exact results as that given by Adomian himself. Also, the modifed simple Adomian polynomial was further tested on concrete problems and the numerical results were exactly the same as the exact solution. The large scale computation and evaluation was made possible by Maple software package . Keywords - Adomian Polynomial, Adomian Decomposition Method. I. I NTRODUCTION
The Adomian Polynomial in ADM has been subject of some studies [1] to [9]. This method generates a solution in form of a series whose terms are determined by a recursive relationship using the Adomian Polynomial. Several authors have suggested different algorithms for computing Adomian Polynomial, prominent among them are [2], [3]. Using the algorithm presented by Adomian himself [1] requires classification of terms in both the ordinary and the accelerated form, which is very complicated for large n (order of the derivative). Algorithm presented by [2] uses Taylor series expansion of the functional which is complicated especially when the unknown appears at the denominator. Calculation of the nth Adomian Polynomials using [3] requires computing the nth order derivative which is complicated for large n. That is why most literatures gives, at most, the first five generated Adomian Polynomial. Despite all the difficulties in applying the used method in [5], it cannot be applied to functionals with several variables. Here we suggest a new simplified single line algorithm that can be implemented in any computer algebraic system. To generate the Adomian Polynomial without resulting to writing codes before implementation
II. THE ADOMIAN POLYNOMIAL IN ADM
Consider the general nonlinear differential equation; fFu (1) F is nonlinear differential operator and u, f are functions of t. Equation (1) in operator form is given as; fNuRuLu (2) where L is an operator representing linear portion of f which is easily invertible, R is a linear operator for the remainder of the linear portion. N is a nonlinear operator representing the nonlinear term in f. Applying the inverse operator L -1 on equation (2) we have; NuLRuLfLLuL (3) By virtue of L, L -1 would represent integration with any given initial/boundary conditions. Equation (3) becomes; NuLRuL)t(g)t(u (4) where g(t) represent the function generated by integrating f and using the initial/boundary conditions. ADM admit the decomposition into an infinite series with equation (4) given as;
0n n10n n100n n
AL)t(RuLu)t(u where A n is the Adomian Polynomial which is given as; )u(Ndd!n1A (5) odified Adomian Polynomial for Nonlinear Functional with Integer Exponent )t(gu (6) n111n ALRuLu (7) Having determined the components u n ; n ≥ 0 the solution
0n n )t(uu (8) is in series form. The series may be summed to provide the solution in closed form. Or, for concrete problems the nth partial sum may be used to give the approximate solution. We give the simple modification to the Adomian Polynomial of equation (5) as;
0n nnnn dd!n1A (9) where n0i i 0j injij )u...uu(... III. IMPLEMENTATION OF THE SIMPLE MODIFIED ADOMIAN POLYNOMIAL
In this section, we present some examples that resulted from the use of the simple modified Adomian polynomial. . For N(u) = and using equation (9), the first ten plus one Adomian Polynomials are given as; uA uu3A uu3uu3A uuuu6uu3A uu3uu3uuu6uu3A uu3uu3uuu3uuu6uu3A uuuu6uu3uu3uuu6uuu6uu3A uu3uu3uuu6uuu6uu3uuu6uuu6uu3A uu3uuu6uu3uuu6uuu6uu3uuu6uuu6uu3A uu3 uu3uuu6uuu6uuu6uuu6uu3uuu6uuu6uu3A uuuu6uu3 uu3uuu6uuu6uuu6uuu6uu3uuu6uuu6uu3A uu3uu3uuu6uuu6uu3 . For N(u) = and using equation (9), the first ten plus one Adomian Polynomials are also given as; uA uu4A uu6uu4A uu4uuu12uu4A odified Adomian Polynomial for Nonlinear Functional with Integer Exponent uuuu12uu6uuu12uu4A uu4uuu12uuu12uuu12uuu12uu4A uu4uu4uuuu24uu6uuu12uuu12uuu12uu4A uu6 uuu12uu4uu4uuu12uuu12uuu12uuu12uu4A uuuu24uuu12uuu12 uu4uuu12uuu12uuuu24uuuu24uuu12uuu4A uuu12uuu12uuu12uuu12uu6uuu12uu6 uuuu24uuu12uuuu24uuu12uuu12uuu12uu4A uuu12uuu12uuu12uuu12uuu12uuu12uuuu24 uu4uu4uu4uuu12 uuuu24uuu12uuuu24uuu12uuu12uuu12uu4A uuu12uuu12uuu12uuu12uuu12uuu12uuuu24 uu4uu4uu4uuu12uuu12uuuu24uuuu24 uu6uu6uu6 IV. APPLICATION OF THE SIMPLE MODIFIED ADOMIAN POLYNOMIAL TO CONCRETE PROBLEMS
In this section, we apply the modified Adomian polynomials to concrete problems.
Problem 1
Consider uu5dtdu , u(0) = 1 (10) The exact solution of equation (10) is given as; t10 (11) And in series form equation (11) is given as t470t100t24t61u (12) Applying the ADM to equation (10), we have; )A(L)u5(L)0(u)t(u n11 where A n in this case is given as u)u(NA Applying equation (6), (7) and (9) to equation (10), we obtain; t0 3001 t6dt)uu5(u t0 213012 t24dt)uu3u5(u t0 321022023 t100dt)uu3uu3u5(u t0 43121032034 t470dt)uuuu6uu3u5(u odified Adomian Polynomial for Nonlinear Functional with Integer Exponent t0 522122031042045 t2336dt)uu3uu3uuu6uu3u5(u t0 622132152041052056 t335588dt)uu3uu3uuu3uuu6uu3u5(u t0 73232123042142051062067 t211282984dt)uuuu6uu3uu3uuu6uuu6uu3u5(u t0 32223142143052152061072078 dt)uu3uu3uuu6uuu6uu3uuu6uuu6uu3u5(u t216681580 Continuing in this order, the sum of the first few terms of u n , is given as; t211282984t335588t2336t470t100t24t61uu . . . This is obviously the same as the series form of the exact solution given in equations (12). The similarity between the exact solution, equation (11) and the numerical solution of the first 12th terms is further given in Fig. 1 and Fig. 2 respectively.
Problem 2
Consider (13) The exact solution of equation (13) is t33 (14) And in series form equation (14) is given as t12701t349t5t21u . . . (15) Applying the Adomian decomposition method to equation (13) we have )A(L)u(L)0(u)t(u n11 where A n in this case is given as; A n = N(u) = u Also, applying the recursive relation and the simple modified Adomian polynomial to equation (13), we obtain; odified Adomian Polynomial for Nonlinear Functional with Integer Exponent t0 4001 t2dt)uu(u t0 213012 t5dt)uu4u(u t0 3212023023 t349dt)uu6uu4u(u t0 4310212033034 t12701dt)uu4uuu12uu4u(u t0 54122102220312043045 t6013081dt)uuuu12uu6uuu12uu4u(u t0 6312221032103220412053056 t7260193dt)uu4uuu12uuu12uuu12uuu12uu4u(u t0 3203313210232042104220512063067 uu4uu4uuuu24uu6uuu12uuu12uuu12uu4u(u t25208231329dt)uu6 Continuing in this order, we have; t25208231329t7260193t6013081t12701t349t5t21uu . . . The first few terms of the series are obviously the same as equation (15) of the exact solution of Problem 2. The resemblance of the numerical solution using the simple modified Adomian polynomial of equation (9) and the exact result is further depicted in Fig. 3 and Fig. 4. In Fig. 2 and Fig. 4, finite terms of the series, uu , were used in the plot. The remarkable similarities between the exact and ADM using equation (9) (the simple modified Adomian polynomial) of problems 1and 2 is further shown in Tables 1 and 2. odified Adomian Polynomial for Nonlinear Functional with Integer Exponent
11 0n n uu -0.14 4.5480323980 x 10 -1 -1 -0.13 4.8432134540 x 10 -1 -1 -0.12 5.1270154680 x 10 -1 -1 -0.11 5.4143250440 x 10 -1 -1 -0.10 5.7131765330 x 10 -1 -1 -1 -1 Table II: Exact versus ADM solution of Probem 2 t Exact solution Solution by ADM,
11 0n n uu -0.14 7.8784892490 x 10 -1 -1 -0.13 7.9983309430 x 10 -1 -1 -0.12 8.1214566950 x 10 -1 -1 -0.11 8.2483167170 x 10 -1 -1 -0.10 8.3792754430 x 10 -1 -1 -1 -1 V. CONCLUSION
In this paper, we proposed an efficient simple modification of the standard Adomian Polynomial in the popular Adomian decomposition method for solving nonlinear functional whose nonlinear term is of the form N(u) = u n . The study showed that the modified Adomian polynomial is simple and is efficient, and also effective in any computer algebra system to get as many term of the Adomian polynomials as required without difficulties. The outcome from the modifications is the same as those presented by Adomian himself. And when applied to concrete problems the results were remarkable. REFERENCES [1]. G. Adomian, A Review of the Decomposition Method in Applied Mathematics,
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