On solutions for several systems of complex nonlinear partial differential equations with two variables
Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existen...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Xu, Hong Yan [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2023 |
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| Schlagwörter: |
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| Anmerkung: |
© The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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| Übergeordnetes Werk: |
Enthalten in: Analysis and Mathematical Physics - Cham (ZG) : Springer International Publishing AG, 2011, 13(2023), 3 vom: 22. Mai |
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| Übergeordnetes Werk: |
volume:13 ; year:2023 ; number:3 ; day:22 ; month:05 |
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| DOI / URN: |
10.1007/s13324-023-00811-z |
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| Katalog-ID: |
SPR051575957 |
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| 245 | 1 | 0 | |a On solutions for several systems of complex nonlinear partial differential equations with two variables |
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| 520 | |a Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations. | ||
| 650 | 4 | |a Nevanlinna theory |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Entire function |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a System of functional equations |7 (dpeaa)DE-He213 | |
| 650 | 4 | |a Several complex variables |7 (dpeaa)DE-He213 | |
| 700 | 1 | |a Xu, Yi Hui |4 aut | |
| 700 | 1 | |a Liu, Xiao Lan |4 aut | |
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10.1007/s13324-023-00811-z doi (DE-627)SPR051575957 (SPR)s13324-023-00811-z-e DE-627 ger DE-627 rakwb eng Xu, Hong Yan verfasserin aut On solutions for several systems of complex nonlinear partial differential equations with two variables 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations. Nevanlinna theory (dpeaa)DE-He213 Entire function (dpeaa)DE-He213 System of functional equations (dpeaa)DE-He213 Several complex variables (dpeaa)DE-He213 Xu, Yi Hui aut Liu, Xiao Lan aut Enthalten in Analysis and Mathematical Physics Cham (ZG) : Springer International Publishing AG, 2011 13(2023), 3 vom: 22. Mai (DE-627)644278323 (DE-600)2588449-9 1664-235X nnns volume:13 year:2023 number:3 day:22 month:05 https://dx.doi.org/10.1007/s13324-023-00811-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 13 2023 3 22 05 |
| spelling |
10.1007/s13324-023-00811-z doi (DE-627)SPR051575957 (SPR)s13324-023-00811-z-e DE-627 ger DE-627 rakwb eng Xu, Hong Yan verfasserin aut On solutions for several systems of complex nonlinear partial differential equations with two variables 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations. Nevanlinna theory (dpeaa)DE-He213 Entire function (dpeaa)DE-He213 System of functional equations (dpeaa)DE-He213 Several complex variables (dpeaa)DE-He213 Xu, Yi Hui aut Liu, Xiao Lan aut Enthalten in Analysis and Mathematical Physics Cham (ZG) : Springer International Publishing AG, 2011 13(2023), 3 vom: 22. Mai (DE-627)644278323 (DE-600)2588449-9 1664-235X nnns volume:13 year:2023 number:3 day:22 month:05 https://dx.doi.org/10.1007/s13324-023-00811-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 13 2023 3 22 05 |
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10.1007/s13324-023-00811-z doi (DE-627)SPR051575957 (SPR)s13324-023-00811-z-e DE-627 ger DE-627 rakwb eng Xu, Hong Yan verfasserin aut On solutions for several systems of complex nonlinear partial differential equations with two variables 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations. Nevanlinna theory (dpeaa)DE-He213 Entire function (dpeaa)DE-He213 System of functional equations (dpeaa)DE-He213 Several complex variables (dpeaa)DE-He213 Xu, Yi Hui aut Liu, Xiao Lan aut Enthalten in Analysis and Mathematical Physics Cham (ZG) : Springer International Publishing AG, 2011 13(2023), 3 vom: 22. Mai (DE-627)644278323 (DE-600)2588449-9 1664-235X nnns volume:13 year:2023 number:3 day:22 month:05 https://dx.doi.org/10.1007/s13324-023-00811-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 13 2023 3 22 05 |
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10.1007/s13324-023-00811-z doi (DE-627)SPR051575957 (SPR)s13324-023-00811-z-e DE-627 ger DE-627 rakwb eng Xu, Hong Yan verfasserin aut On solutions for several systems of complex nonlinear partial differential equations with two variables 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations. Nevanlinna theory (dpeaa)DE-He213 Entire function (dpeaa)DE-He213 System of functional equations (dpeaa)DE-He213 Several complex variables (dpeaa)DE-He213 Xu, Yi Hui aut Liu, Xiao Lan aut Enthalten in Analysis and Mathematical Physics Cham (ZG) : Springer International Publishing AG, 2011 13(2023), 3 vom: 22. Mai (DE-627)644278323 (DE-600)2588449-9 1664-235X nnns volume:13 year:2023 number:3 day:22 month:05 https://dx.doi.org/10.1007/s13324-023-00811-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 13 2023 3 22 05 |
| allfieldsSound |
10.1007/s13324-023-00811-z doi (DE-627)SPR051575957 (SPR)s13324-023-00811-z-e DE-627 ger DE-627 rakwb eng Xu, Hong Yan verfasserin aut On solutions for several systems of complex nonlinear partial differential equations with two variables 2023 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations. Nevanlinna theory (dpeaa)DE-He213 Entire function (dpeaa)DE-He213 System of functional equations (dpeaa)DE-He213 Several complex variables (dpeaa)DE-He213 Xu, Yi Hui aut Liu, Xiao Lan aut Enthalten in Analysis and Mathematical Physics Cham (ZG) : Springer International Publishing AG, 2011 13(2023), 3 vom: 22. Mai (DE-627)644278323 (DE-600)2588449-9 1664-235X nnns volume:13 year:2023 number:3 day:22 month:05 https://dx.doi.org/10.1007/s13324-023-00811-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_65 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_101 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4328 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 AR 13 2023 3 22 05 |
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English |
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Enthalten in Analysis and Mathematical Physics 13(2023), 3 vom: 22. Mai volume:13 year:2023 number:3 day:22 month:05 |
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Enthalten in Analysis and Mathematical Physics 13(2023), 3 vom: 22. Mai volume:13 year:2023 number:3 day:22 month:05 |
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Xu, Hong Yan @@aut@@ Xu, Yi Hui @@aut@@ Liu, Xiao Lan @@aut@@ |
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Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Nevanlinna theory</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Entire function</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">System of functional equations</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Several complex variables</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Xu, Yi Hui</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Liu, Xiao Lan</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Analysis and Mathematical Physics</subfield><subfield code="d">Cham (ZG) : Springer International Publishing AG, 2011</subfield><subfield code="g">13(2023), 3 vom: 22. 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Xu, Hong Yan |
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Xu, Hong Yan misc Nevanlinna theory misc Entire function misc System of functional equations misc Several complex variables On solutions for several systems of complex nonlinear partial differential equations with two variables |
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On solutions for several systems of complex nonlinear partial differential equations with two variables Nevanlinna theory (dpeaa)DE-He213 Entire function (dpeaa)DE-He213 System of functional equations (dpeaa)DE-He213 Several complex variables (dpeaa)DE-He213 |
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misc Nevanlinna theory misc Entire function misc System of functional equations misc Several complex variables |
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misc Nevanlinna theory misc Entire function misc System of functional equations misc Several complex variables |
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misc Nevanlinna theory misc Entire function misc System of functional equations misc Several complex variables |
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Elektronische Aufsätze Aufsätze Elektronische Ressource |
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On solutions for several systems of complex nonlinear partial differential equations with two variables |
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On solutions for several systems of complex nonlinear partial differential equations with two variables |
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Xu, Hong Yan |
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Analysis and Mathematical Physics |
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Xu, Hong Yan |
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10.1007/s13324-023-00811-z |
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on solutions for several systems of complex nonlinear partial differential equations with two variables |
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On solutions for several systems of complex nonlinear partial differential equations with two variables |
| abstract |
Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| abstractGer |
Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
| abstract_unstemmed |
Abstract This article is devoted to describe the entire solutions of several systems of the first order nonlinear partial differential equations. By using the Nevanlinna theory and the Hadamard factorization theory of meromorphic functions, we establish some interesting results to reveal the existence and the forms of the finite order transcendental entire solutions of several systems of the first order nonlinear partial differential equations auz1+buz2cvz1+dvz2=eg,avz1+bvz2cuz1+duz2=eg,auz1+bvz2cuz2+dvz1=eg,auz2+bvz1cuz1+dvz2=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bu_{z_2}\right) \left( cv_{z_1}+dv_{z_2}\right) =e^g,\\&\left( av_{z_1}+bv_{z_2}\right) \left( cu_{z_1}+d u_{z_2}\right) =e^g, \end{aligned} \right. \\ \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_2}\right) \left( cu_{z_2}+dv_{z_1}\right) =e^g,\\&\left( au_{z_2}+bv_{z_1}\right) \left( cu_{z_1}+dv_{z_2}\right) =e^g, \end{aligned} \right. \end{aligned}%$and auz1+bvz1cuz2+dvz2=eg,auz2+bvz2cuz1+dvz1=eg,%$\begin{aligned} \left\{ \begin{aligned}&\left( au_{z_1}+bv_{z_1}\right) \left( cu_{z_2}+dv_{z_2}\right) =e^g,\\&\left( au_{z_2}+bv_{z_2}\right) \left( cu_{z_1}+dv_{z_1}\right) =e^g, \end{aligned} \right. \end{aligned}%$where %$a,b,c,d\in {\mathbb {C}}%$, and g is a polynomial in %${\mathbb {C}}^2%$. Moreover, some examples are given to explain that there are significant differences in the forms of solutions from some previous systems of functional equations. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law. |
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On solutions for several systems of complex nonlinear partial differential equations with two variables |
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https://dx.doi.org/10.1007/s13324-023-00811-z |
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Xu, Yi Hui Liu, Xiao Lan |
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| doi_str |
10.1007/s13324-023-00811-z |
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2024-07-03T22:37:08.531Z |
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| score |
7.3998966 |