Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions
This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct m...
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| Format: | Article |
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MDPI AG
2021-02-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/4/345 |
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| author | Alexander V. Aksenov Andrei D. Polyanin |
| author_facet | Alexander V. Aksenov Andrei D. Polyanin |
| author_sort | Alexander V. Aksenov |
| collection | DOAJ |
| description | This paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, these equations contain the same function at a past time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, also contain the same functions with dilated or contracted arguments, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>p</mi><mi>x</mi><mo>,</mo><mi>q</mi><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>p</i> and <i>q</i> are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”. |
| first_indexed | 2024-03-09T04:59:13Z |
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| id | doaj.art-899c478864364022ba750c5cd7e86794 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T04:59:13Z |
| publishDate | 2021-02-01 |
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| spelling | doaj.art-899c478864364022ba750c5cd7e867942023-12-03T13:01:37ZengMDPI AGMathematics2227-73902021-02-019434510.3390/math9040345Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler SolutionsAlexander V. Aksenov0Andrei D. Polyanin1Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, 1 Leninskiye Gory, Main Building, 119234 Moscow, RussiaIshlinsky Institute for Problems in Mechanics RAS, 101 Vernadsky Avenue, Bldg 1, 119526 Moscow, RussiaThis paper describes a number of simple but quite effective methods for constructing exact solutions of nonlinear partial differential equations that involve a relatively small amount of intermediate calculations. The methods employ two main ideas: (i) simple exact solutions can serve to construct more complex solutions of the equations under consideration and (ii) exact solutions of some equations can serve to construct solutions of other, more complex equations. In particular, we propose a method for constructing complex solutions from simple solutions using translation and scaling. We show that in some cases, rather complex solutions can be obtained by adding one or more terms to simpler solutions. There are situations where nonlinear superposition allows us to construct a complex composite solution using similar simple solutions. We also propose a few methods for constructing complex exact solutions to linear and nonlinear PDEs by introducing complex-valued parameters into simpler solutions. The effectiveness of the methods is illustrated by a large number of specific examples (over 30 in total). These include nonlinear heat equations, reaction–diffusion equations, wave type equations, Klein–Gordon type equations, equations of motion through porous media, hydrodynamic boundary layer equations, equations of motion of a liquid film, equations of gas dynamics, Navier–Stokes equations, and some other PDEs. Apart from exact solutions to ‘ordinary’ partial differential equations, we also describe some exact solutions to more complex nonlinear delay PDEs. Along with the unknown function at the current time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, these equations contain the same function at a past time, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula> is the delay time. Furthermore, we look at nonlinear partial functional-differential equations of the pantograph type, which, in addition to the unknown <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>u</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, also contain the same functions with dilated or contracted arguments, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>w</mi><mo>=</mo><mi>u</mi><mo>(</mo><mi>p</mi><mi>x</mi><mo>,</mo><mi>q</mi><mi>t</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>p</i> and <i>q</i> are scaling parameters. We propose an efficient approach to construct exact solutions to such functional-differential equations. Some new exact solutions of nonlinear pantograph-type PDEs are presented. The methods and examples in this paper are presented according to the principle “from simple to complex”.https://www.mdpi.com/2227-7390/9/4/345exact solutionsnonlinear PDEsreaction–diffusion equationswave type equationsPDEs with constant and variable delaypantograph-type PDEs |
| spellingShingle | Alexander V. Aksenov Andrei D. Polyanin Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions Mathematics exact solutions nonlinear PDEs reaction–diffusion equations wave type equations PDEs with constant and variable delay pantograph-type PDEs |
| title | Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions |
| title_full | Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions |
| title_fullStr | Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions |
| title_full_unstemmed | Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions |
| title_short | Methods for Constructing Complex Solutions of Nonlinear PDEs Using Simpler Solutions |
| title_sort | methods for constructing complex solutions of nonlinear pdes using simpler solutions |
| topic | exact solutions nonlinear PDEs reaction–diffusion equations wave type equations PDEs with constant and variable delay pantograph-type PDEs |
| url | https://www.mdpi.com/2227-7390/9/4/345 |
| work_keys_str_mv | AT alexandervaksenov methodsforconstructingcomplexsolutionsofnonlinearpdesusingsimplersolutions AT andreidpolyanin methodsforconstructingcomplexsolutionsofnonlinearpdesusingsimplersolutions |