Integration of Dissipative Partial Differential Equations: A Case Study

2010 ◽  
Vol 9 (3) ◽  
pp. 1119-1133 ◽  
Author(s):  
Gianni Arioli ◽  
Hans Koch
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Dissipative Partial Differential Equations

On the crest factor for dissipative partial differential equations

2019 ◽  
Vol 475 (2229) ◽  
pp. 20190322
Author(s):  
Michele V. Bartuccelli
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Positive Parameter ◽  
Crest Factor ◽  
Partial Differential ◽  
Large Potential ◽  
Dissipative Partial Differential Equations

In this work, we have introduced and then computed the so-called crest factor associated with solutions of dissipative partial differential equations (PDEs). By taking two paradigmatic dissipative PDEs, we estimated in an explicit and accurate manner the values of the crest factor of their solutions. We then analysed and compared the estimates as a function of the positive parameter which appears in the PDEs in space dimensions one and two. These estimates shed some light on the dynamics of the fluctuations of the solutions of the two model PDEs, and therefore provide a criterion for discerning between small and large potential excursions in space for the solution of any dissipative PDE. Being able to detect between small and large intermittent fluctuations is one of the hallmarks of turbulence. We believe that the crest factor is an appropriate tool for extracting space fluctuation features in solutions of dissipative PDEs.

scholarly journals Improvement of the Modified Decomposition Method for Handling Third-Order Singular Nonlinear Partial Differential Equations with Applications in Physics

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Nemat Dalir
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Decomposition Method ◽  
Differential Operators ◽  
Nonlinear Partial Differential Equations ◽  
Initial Value ◽  
Third Order ◽  
Partial Differential ◽  
The Third

The modified decomposition method (MDM) is improved by introducing new inverse differential operators to adapt the MDM for handling third-order singular nonlinear partial differential equations (PDEs) arising in physics and mechanics. A few case-study singular nonlinear initial-value problems (IVPs) of third-order PDEs are presented and solved by the improved modified decomposition method (IMDM). The solutions are compared with the existing exact analytical solutions. The comparisons show that the IMDM is effectively capable of obtaining the exact solutions of the third-order singular nonlinear IVPs.

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