scholarly journals A direct method to find Stokes multipliers in closed form for P$_1$ and more general integrable systems

2015 ◽  
Vol 368 (11) ◽  
pp. 7579-7621 ◽  
Author(s):  
O. Costin ◽  
R. D. Costin ◽  
M. Huang
Keyword(s):  
Closed Form ◽  
Integrable Systems ◽  
Direct Method ◽  
A Direct Method

scholarly journals New Conditions for Obtaining the Exact Solutions of the General Riccati Equation

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Lazhar Bougoffa
Keyword(s):  
Exact Solutions ◽  
Closed Form ◽  
Riccati Equation ◽  
Direct Method ◽  
Equivalent Equation ◽  
A Direct Method

We propose a direct method for solving the general Riccati equationy′=f(x)+g(x)y+h(x)y2. We first reduce it into an equivalent equation, and then we formulate the relations between the coefficients functionsf(x),g(x), andh(x)of the equation to obtain an equivalent separable equation from which the previous equation can be solved in closed form. Several examples are presented to demonstrate the efficiency of this method.

A Note on the Use of Approximate Solutions for the Bending Vibrations of Simply Supported Cracked Beams

2010 ◽  
Vol 132 (2) ◽  
Author(s):  
L. Rubio ◽  
J. Fernández-Sáez
Keyword(s):  
Closed Form ◽  
Natural Frequencies ◽  
Direct Method ◽  
Approximate Solutions ◽  
Cracked Beam ◽  
Bending Vibrations ◽  
Closed Form Solutions ◽  
Simply Supported ◽  
A Direct Method ◽  
Cracked Beams

The main goal of this note is to discuss the applicability of approximate closed-form solutions to evaluate the natural frequencies for bending vibrations of simply supported Euler–Bernoulli cracked beams. From the well-known model, which considers the cracked beam as two beams connected by a rotational spring, different approximate solutions are revisited and compared with those found by a direct method, which has been chosen as reference.

scholarly journals Plastic waste to fuels by hydrocracking at mild conditions

2021 ◽  
Vol 7 (17) ◽  
pp. eabf8283
Author(s):  
Sibao Liu ◽  
Pavel A. Kots ◽  
Brandon C. Vance ◽  
Andrew Danielson ◽  
Dionisios G. Vlachos
Keyword(s):  
Direct Method ◽  
Acid Sites ◽  
Liquid Fuels ◽  
High Yield ◽  
Tandem Catalysis ◽  
Hy Zeolite ◽  
Environmental Threat ◽  
Single Use ◽  
Initial Activation ◽  
A Direct Method

Single-use plastics impose an enormous environmental threat, but their recycling, especially of polyolefins, has been proven challenging. We report a direct method to selectively convert polyolefins to branched, liquid fuels including diesel, jet, and gasoline-range hydrocarbons, with high yield up to 85% over Pt/WO3/ZrO2 and HY zeolite in hydrogen at temperatures as low as 225°C. The process proceeds via tandem catalysis with initial activation of the polymer primarily over Pt, with subsequent cracking over the acid sites of WO3/ZrO2 and HY zeolite, isomerization over WO3/ZrO2 sites, and hydrogenation of olefin intermediates over Pt. The process can be tuned to convert different common plastic wastes, including low- and high-density polyethylene, polypropylene, polystyrene, everyday polyethylene bottles and bags, and composite plastics to desirable fuels and light lubricants.

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