scholarly journals A mixed problem for hyperbolic equations of second order with a first order derivative boundary condition

1969 ◽  
Vol 5 (2) ◽  
pp. 119-147 ◽  
Author(s):  
Mitsuru Ikawa
Keyword(s):  
Boundary Condition ◽  
Mixed Problem ◽  
Hyperbolic Equations ◽  
Second Order ◽  
Order Derivative ◽  
First Order

Chaotic Oscillations of Second Order Linear Hyperbolic Equations with Nonlinear Boundary Conditions: A Factorizable but Noncommutative Case

2015 ◽  
Vol 25 (11) ◽  
pp. 1530032 ◽  
Author(s):  
Liangliang Li ◽  
Yu Huang ◽  
Goong Chen ◽  
Tingwen Huang
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Hyperbolic Equations ◽  
Nonlinear Boundary Conditions ◽  
Second Order ◽  
Chaotic Oscillations ◽  
Van Der Pol ◽  
Order Linear ◽  
First Order

If a second order linear hyperbolic partial differential equation in one-space dimension can be factorized as a product of two first order operators and if the two first order operators commute, with one boundary condition being the van der Pol type and the other being linear, one can establish the occurrence of chaos when the parameters enter a certain regime [Chen et al., 2014]. However, if the commutativity of the two first order operators fails to hold, then the treatment in [Chen et al., 2014] no longer works and significant new challenges arise in determining nonlinear boundary conditions that engenders chaos. In this paper, we show that by incorporating a linear memory effect, a nonlinear van der Pol boundary condition can cause chaotic oscillations when the parameter enters a certain regime. Numerical simulations illustrating chaotic oscillations are also presented.

scholarly journals Estimation of Tadalafil Using Derivative Spectrophotometry in Bulk Material and in Pharmaceutical Formulation

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Zamir G. Khan ◽  
Amod S. Patil ◽  
Atul A. Shirkhedkar
Keyword(s):  
Wavelength Range ◽  
Area Under Curve ◽  
Pharmaceutical Formulation ◽  
Second Order ◽  
Order Derivative ◽  
Uv Spectrophotometry ◽  
First Order ◽  
Beer's Law ◽  
Beer’S Law ◽  
Derivative Uv Spectrophotometry

Four simple, rapid, accurate, precise, reliable, and economical UV-spectrophotometric methods have been proposed for the determination of tadalafil in bulk and in pharmaceutical formulation. “Method A” is first order derivative UV spectrophotometry using amplitude, “method B” is first order derivative UV spectrophotometry using area under curve technique, “method C” is second order derivative UV spectrophotometry using amplitude, and “method D” is second order derivative UV spectrophotometry using area under curve technique. The developed methods have shown best results in terms of linearity, accuracy, precision, and LOD and LOQ for bulk drug and marketed formulation as well. In N,N-dimethylformamide, tadalafil showed maximum absorbance at 284 nm. For “method A” amplitude was recorded at 297 nm while for “method B” area under curve was integrated in the wavelength range of 290.60–304.40 nm. For “method C” amplitude was measured at 284 nm while for “method D” area under curve was selected in the wavelength range of 280.80–286.20 nm. For methods A and B, tadalafil obeyed Lambert-Beer’s law in the range of 05–50 μg/mL while for “methods C and D”, tadalafil obeyed Lambert-Beer’s law in the range of 20–70 μg/mL, and-for “methods A, B, C, and D” the correlation coefficients were found to be > than 0.999.

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