A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations
A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Kutniv, M.V. [verfasserIn] Datsko, B.Y. [verfasserIn] Kunynets, A.V. [verfasserIn] Włoch, A. [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2019 |
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| Schlagwörter: |
Nonlinear ordinary differential equation |
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| Übergeordnetes Werk: |
Enthalten in: Applied numerical mathematics - Amsterdam [u.a.] : Elsevier, 1985, 148, Seite 140-151 |
|---|---|
| Übergeordnetes Werk: |
volume:148 ; pages:140-151 |
| DOI / URN: |
10.1016/j.apnum.2019.09.006 |
|---|
| Katalog-ID: |
ELV002973286 |
|---|
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| 024 | 7 | |a 10.1016/j.apnum.2019.09.006 |2 doi | |
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| 100 | 1 | |a Kutniv, M.V. |e verfasserin |4 aut | |
| 245 | 1 | 0 | |a A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations |
| 264 | 1 | |c 2019 | |
| 336 | |a nicht spezifiziert |b zzz |2 rdacontent | ||
| 337 | |a Computermedien |b c |2 rdamedia | ||
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| 520 | |a A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. | ||
| 650 | 4 | |a Nonlinear ordinary differential equation | |
| 650 | 4 | |a Singular initial value problem | |
| 650 | 4 | |a Taylor series method | |
| 650 | 4 | |a Runge-Kutta methods | |
| 700 | 1 | |a Datsko, B.Y. |e verfasserin |4 aut | |
| 700 | 1 | |a Kunynets, A.V. |e verfasserin |4 aut | |
| 700 | 1 | |a Włoch, A. |e verfasserin |4 aut | |
| 773 | 0 | 8 | |i Enthalten in |t Applied numerical mathematics |d Amsterdam [u.a.] : Elsevier, 1985 |g 148, Seite 140-151 |h Online-Ressource |w (DE-627)266888879 |w (DE-600)1468770-7 |w (DE-576)075962314 |7 nnns |
| 773 | 1 | 8 | |g volume:148 |g pages:140-151 |
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| 936 | b | k | |a 31.76 |j Numerische Mathematik |
| 951 | |a AR | ||
| 952 | |d 148 |h 140-151 | ||
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10.1016/j.apnum.2019.09.006 doi (DE-627)ELV002973286 (ELSEVIER)S0168-9274(19)30245-4 DE-627 ger DE-627 rda eng 510 DE-600 31.76 bkl Kutniv, M.V. verfasserin aut A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations 2019 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. Nonlinear ordinary differential equation Singular initial value problem Taylor series method Runge-Kutta methods Datsko, B.Y. verfasserin aut Kunynets, A.V. verfasserin aut Włoch, A. verfasserin aut Enthalten in Applied numerical mathematics Amsterdam [u.a.] : Elsevier, 1985 148, Seite 140-151 Online-Ressource (DE-627)266888879 (DE-600)1468770-7 (DE-576)075962314 nnns volume:148 pages:140-151 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4393 31.76 Numerische Mathematik AR 148 140-151 |
| spelling |
10.1016/j.apnum.2019.09.006 doi (DE-627)ELV002973286 (ELSEVIER)S0168-9274(19)30245-4 DE-627 ger DE-627 rda eng 510 DE-600 31.76 bkl Kutniv, M.V. verfasserin aut A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations 2019 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. Nonlinear ordinary differential equation Singular initial value problem Taylor series method Runge-Kutta methods Datsko, B.Y. verfasserin aut Kunynets, A.V. verfasserin aut Włoch, A. verfasserin aut Enthalten in Applied numerical mathematics Amsterdam [u.a.] : Elsevier, 1985 148, Seite 140-151 Online-Ressource (DE-627)266888879 (DE-600)1468770-7 (DE-576)075962314 nnns volume:148 pages:140-151 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4393 31.76 Numerische Mathematik AR 148 140-151 |
| allfields_unstemmed |
10.1016/j.apnum.2019.09.006 doi (DE-627)ELV002973286 (ELSEVIER)S0168-9274(19)30245-4 DE-627 ger DE-627 rda eng 510 DE-600 31.76 bkl Kutniv, M.V. verfasserin aut A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations 2019 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. Nonlinear ordinary differential equation Singular initial value problem Taylor series method Runge-Kutta methods Datsko, B.Y. verfasserin aut Kunynets, A.V. verfasserin aut Włoch, A. verfasserin aut Enthalten in Applied numerical mathematics Amsterdam [u.a.] : Elsevier, 1985 148, Seite 140-151 Online-Ressource (DE-627)266888879 (DE-600)1468770-7 (DE-576)075962314 nnns volume:148 pages:140-151 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4393 31.76 Numerische Mathematik AR 148 140-151 |
| allfieldsGer |
10.1016/j.apnum.2019.09.006 doi (DE-627)ELV002973286 (ELSEVIER)S0168-9274(19)30245-4 DE-627 ger DE-627 rda eng 510 DE-600 31.76 bkl Kutniv, M.V. verfasserin aut A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations 2019 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. Nonlinear ordinary differential equation Singular initial value problem Taylor series method Runge-Kutta methods Datsko, B.Y. verfasserin aut Kunynets, A.V. verfasserin aut Włoch, A. verfasserin aut Enthalten in Applied numerical mathematics Amsterdam [u.a.] : Elsevier, 1985 148, Seite 140-151 Online-Ressource (DE-627)266888879 (DE-600)1468770-7 (DE-576)075962314 nnns volume:148 pages:140-151 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4393 31.76 Numerische Mathematik AR 148 140-151 |
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10.1016/j.apnum.2019.09.006 doi (DE-627)ELV002973286 (ELSEVIER)S0168-9274(19)30245-4 DE-627 ger DE-627 rda eng 510 DE-600 31.76 bkl Kutniv, M.V. verfasserin aut A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations 2019 nicht spezifiziert zzz rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. Nonlinear ordinary differential equation Singular initial value problem Taylor series method Runge-Kutta methods Datsko, B.Y. verfasserin aut Kunynets, A.V. verfasserin aut Włoch, A. verfasserin aut Enthalten in Applied numerical mathematics Amsterdam [u.a.] : Elsevier, 1985 148, Seite 140-151 Online-Ressource (DE-627)266888879 (DE-600)1468770-7 (DE-576)075962314 nnns volume:148 pages:140-151 GBV_USEFLAG_U SYSFLAG_U GBV_ELV SSG-OPC-MAT GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_150 GBV_ILN_151 GBV_ILN_224 GBV_ILN_370 GBV_ILN_602 GBV_ILN_702 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2027 GBV_ILN_2034 GBV_ILN_2038 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2056 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2190 GBV_ILN_2336 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4313 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4338 GBV_ILN_4393 31.76 Numerische Mathematik AR 148 140-151 |
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Kutniv, M.V. |
| doi_str_mv |
10.1016/j.apnum.2019.09.006 |
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| title_sort |
a new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations |
| title_auth |
A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations |
| abstract |
A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. |
| abstractGer |
A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. |
| abstract_unstemmed |
A new approach to construction of one-step numerical methods of high order for the initial value problems on the interval [ 0 , a ] with a singularity of the first kind in the point x = 0 is proposed. Using the substitution of the independent variable x = e t , we reduce the original initial value problem to the one on the interval ( − ∞ , ln a ] . On some finite irregular grid { t n ∈ ( − ∞ , ln a ] , n = 0 , 1 , . . . , N , t N = ln a } Taylor series and Runge-Kutta methods for this problem have been developed. For finding of an approximate solution at the grid node t 0 , new one-step methods have been constructed. For finding of the solution at other grid nodes, the standard one-step methods have been used. An algorithm for the automatic generation of a grid which guarantees the prescribed accuracy is presented. The effectiveness of presented approach is illustrated by a set of numerical examples. The applicability of the constructed method to systems of singular differential equations is shown. |
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| title_short |
A new approach to constructing of explicit one-step methods of high order for singular initial value problems for nonlinear ordinary differential equations |
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| author2 |
Datsko, B.Y. Kunynets, A.V. Włoch, A. |
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10.1016/j.apnum.2019.09.006 |
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2024-07-06T18:04:19.816Z |
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