Symbolic implementation of the algorithm for calculating Adomian polynomials
In this paper, a symbolic implementation code is developed of a technique proposed by Wazwaz [Appl. Math. Comput. 111 (2000) 53] for calculating Adomian polynomials for nonlinear operators. The algorithm proposed by him [Appl. Math. Comput. 111 (2000) 53] offers a promising approach for calculating...
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| Published in: | Applied mathematics and computation Vol. 146; no. 1; pp. 257 - 271 |
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| Format: | Journal Article |
| Language: | English |
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Elsevier Inc
30.12.2003
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| ISSN: | 0096-3003, 1873-5649 |
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| Abstract | In this paper, a symbolic implementation code is developed of a technique proposed by Wazwaz [Appl. Math. Comput. 111 (2000) 53] for calculating Adomian polynomials for nonlinear operators. The algorithm proposed by him [Appl. Math. Comput. 111 (2000) 53] offers a promising approach for calculating Adomian polynomials for all forms of nonlinearity, but it is not easy to implement due to its huge size of algebraic calculations, complicated trigonometric terms, and unique summation rules. It is well known that the algebraic manipulation language such as Mathematica is useful to facilitate such a hard computational work. Pattern-matching capabilities peculiar feature of Mathematica are used in index regrouping which is a key role in constructing Adomian polynomials. The computer algebra software Mathematica is used to collect terms to their order and to simplify the terms. The symbolic implementation code author developed (appearing at appendix) has the flexibility that may easily cover any length of Adomian polynomial for many forms of nonlinear cases. A nonlinear evolution equation is investigated in order to justify the availability of symbolic implementation code. |
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| AbstractList | In this paper, a symbolic implementation code is developed of a technique proposed by Wazwaz [Appl. Math. Comput. 111 (2000) 53] for calculating Adomian polynomials for nonlinear operators. The algorithm proposed by him [Appl. Math. Comput. 111 (2000) 53] offers a promising approach for calculating Adomian polynomials for all forms of nonlinearity, but it is not easy to implement due to its huge size of algebraic calculations, complicated trigonometric terms, and unique summation rules. It is well known that the algebraic manipulation language such as Mathematica is useful to facilitate such a hard computational work. Pattern-matching capabilities peculiar feature of Mathematica are used in index regrouping which is a key role in constructing Adomian polynomials. The computer algebra software Mathematica is used to collect terms to their order and to simplify the terms. The symbolic implementation code author developed (appearing at appendix) has the flexibility that may easily cover any length of Adomian polynomial for many forms of nonlinear cases. A nonlinear evolution equation is investigated in order to justify the availability of symbolic implementation code. |
| Author | Shin, J.-G. Choi, H.-W. |
| Author_xml | – sequence: 1 givenname: H.-W. surname: Choi fullname: Choi, H.-W. email: heydaniel69@hanmail.net organization: Department of Intelligent Machines Research, Hyundai Heavy Industries Electro-Mechanical Research Institute, 102-18, Mabook-Ri, Kuseong-Myun, Yongin-Shi, Kyonggi-Do 449-716, South Korea – sequence: 2 givenname: J.-G. surname: Shin fullname: Shin, J.-G. organization: Department of Naval Architecture and Ocean Engineering, Seoul National University, San 56-1, Seoul 151-742, South Korea |
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| Cites_doi | 10.1016/0022-247X(84)90182-3 10.1016/S0096-3003(96)00281-0 10.1016/S0096-3003(99)00063-6 10.1016/0096-3003(94)00137-S 10.1016/0895-7177(96)00080-5 10.1016/0022-247X(88)90170-9 10.1016/0022-247X(84)90181-1 |
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| Keywords | Mathematica Nonlinear evolution equation Adomian polynomials Adomian decomposition method Pattern matching Nonlinear operators Non linear equation Adomian polynomial Applied mathematics Decomposition method Evolution equation Computer algebra Adomian method Non linear operator |
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| References | Rach (BIB4) 1984; 102 Adomian (BIB1) 1994 Seng, Abbaoui, Cherruault (BIB5) 1996; 24 Abbaoui, Cherruault (BIB6) 1999; 102 Wazwaz (BIB9) 1997; 87 Adomian (BIB3) 1984; 102 Adomian (BIB2) 1988; 135 Wazwaz (BIB7) 1995; 69 Wazwaz (BIB10) 2000; 111 Wazwaz (BIB8) 1997 Rach (10.1016/S0096-3003(02)00541-6_BIB4) 1984; 102 Adomian (10.1016/S0096-3003(02)00541-6_BIB2) 1988; 135 Wazwaz (10.1016/S0096-3003(02)00541-6_BIB7) 1995; 69 Seng (10.1016/S0096-3003(02)00541-6_BIB5) 1996; 24 Wazwaz (10.1016/S0096-3003(02)00541-6_BIB10) 2000; 111 Abbaoui (10.1016/S0096-3003(02)00541-6_BIB6) 1999; 102 Wazwaz (10.1016/S0096-3003(02)00541-6_BIB9) 1997; 87 Adomian (10.1016/S0096-3003(02)00541-6_BIB1) 1994 Adomian (10.1016/S0096-3003(02)00541-6_BIB3) 1984; 102 Wazwaz (10.1016/S0096-3003(02)00541-6_BIB8) 1997 |
| References_xml | – volume: 87 start-page: 199 year: 1997 end-page: 204 ident: BIB9 article-title: A study on a boundary-layer equation arising in an incompressible fluid publication-title: Appl. Math. Comput. – volume: 102 start-page: 420 year: 1984 end-page: 434 ident: BIB3 article-title: A new approach to nonlinear partial differential equations publication-title: J. Math. Anal. Appl. – year: 1994 ident: BIB1 article-title: Solving Frontier Problems of Physics. The Decomposition Method – volume: 102 start-page: 77 year: 1999 end-page: 86 ident: BIB6 article-title: Convergence of Adomian’s method applied to differential equations publication-title: Comput. Math. Appl. – volume: 69 start-page: 299 year: 1995 end-page: 311 ident: BIB7 article-title: The decomposition method for approximate solution of the Goursat problem publication-title: Appl. Math. Comput. – volume: 102 start-page: 415 year: 1984 end-page: 419 ident: BIB4 article-title: A convenient computational form for the Adomian polynomials publication-title: J. Math. Anal. Appl. – volume: 111 start-page: 53 year: 2000 end-page: 69 ident: BIB10 article-title: A new algorithm for calculating adomian polynomials for nonlinear operators publication-title: Appl. Math. Comput. – volume: 135 start-page: 501 year: 1988 end-page: 544 ident: BIB2 article-title: A review of the decomposition method in applied mathematics publication-title: J. Math. Anal. Appl. – volume: 24 start-page: 59 year: 1996 end-page: 65 ident: BIB5 article-title: Adomian’s polynomials for nonlinear operators publication-title: Math. Comput. Modell. – year: 1997 ident: BIB8 article-title: A First Course in Integral Equations – volume: 102 start-page: 420 year: 1984 ident: 10.1016/S0096-3003(02)00541-6_BIB3 article-title: A new approach to nonlinear partial differential equations publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(84)90182-3 – volume: 87 start-page: 199 year: 1997 ident: 10.1016/S0096-3003(02)00541-6_BIB9 article-title: A study on a boundary-layer equation arising in an incompressible fluid publication-title: Appl. Math. Comput. doi: 10.1016/S0096-3003(96)00281-0 – volume: 102 start-page: 77 year: 1999 ident: 10.1016/S0096-3003(02)00541-6_BIB6 article-title: Convergence of Adomian’s method applied to differential equations publication-title: Comput. Math. Appl. – volume: 111 start-page: 53 year: 2000 ident: 10.1016/S0096-3003(02)00541-6_BIB10 article-title: A new algorithm for calculating adomian polynomials for nonlinear operators publication-title: Appl. Math. Comput. doi: 10.1016/S0096-3003(99)00063-6 – volume: 69 start-page: 299 year: 1995 ident: 10.1016/S0096-3003(02)00541-6_BIB7 article-title: The decomposition method for approximate solution of the Goursat problem publication-title: Appl. Math. Comput. doi: 10.1016/0096-3003(94)00137-S – year: 1997 ident: 10.1016/S0096-3003(02)00541-6_BIB8 – volume: 24 start-page: 59 year: 1996 ident: 10.1016/S0096-3003(02)00541-6_BIB5 article-title: Adomian’s polynomials for nonlinear operators publication-title: Math. Comput. Modell. doi: 10.1016/0895-7177(96)00080-5 – year: 1994 ident: 10.1016/S0096-3003(02)00541-6_BIB1 – volume: 135 start-page: 501 year: 1988 ident: 10.1016/S0096-3003(02)00541-6_BIB2 article-title: A review of the decomposition method in applied mathematics publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(88)90170-9 – volume: 102 start-page: 415 year: 1984 ident: 10.1016/S0096-3003(02)00541-6_BIB4 article-title: A convenient computational form for the Adomian polynomials publication-title: J. Math. Anal. Appl. doi: 10.1016/0022-247X(84)90181-1 |
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| SubjectTerms | Adomian decomposition method Adomian polynomials Algorithmics. Computability. Computer arithmetics Applied sciences Artificial intelligence Computer science; control theory; systems Exact sciences and technology Mathematica Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) Nonlinear evolution equation Nonlinear operators Numerical analysis. Scientific computation Pattern matching Pattern recognition. Digital image processing. Computational geometry Sciences and techniques of general use Theoretical computing |
| Title | Symbolic implementation of the algorithm for calculating Adomian polynomials |
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