A direct method of solution of linear differential pursuit-evasion games

1983 ◽  
Vol 33 (6) ◽  
pp. 455-458 ◽  
Author(s):  
M. S. Nikol'skii
Keyword(s):  
Direct Method ◽  
Pursuit Evasion ◽  
Differential Pursuit ◽  
Method Of Solution ◽  
Linear Differential ◽  
A Direct Method

On modeling column crystallizers and a Hermite predictor–corrector scheme for a system of integro-differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Khaldoun El Khaldi ◽  
Nima Rabiei ◽  
Elias G. Saleeby
Keyword(s):  
Numerical Experiments ◽  
Main Part ◽  
Direct Method ◽  
Particle Agglomeration ◽  
Volterra Type ◽  
Type Method ◽  
Method Of Solution ◽  
System Of Integrodifferential Equations ◽  
Predictor Corrector ◽  
A Direct Method

Abstract Multistaged crystallization systems are used in the production of many chemicals. In this article, employing the population balance framework, we develop a model for a column crystallizer where particle agglomeration is a significant growth mechanism. The main part of the model can be reduced to a system of integrodifferential equations (IDEs) of the Volterra type. To solve this system simultaneously, we examine two numerical schemes that yield a direct method of solution and an implicit Runge–Kutta type method. Our numerical experiments show that the extension of a Hermite predictor–corrector method originally advanced in Khanh (1994) for a single IDE is effective in solving our model. The numerical method is presented for a generalization of the model which can be used to study and simulate a number of possible operating profiles of the column.

To Investigate the Euler Solution of Differential Equation

2014 ◽  
Vol 614 ◽  
pp. 409-412
Author(s):  
Lin Long Zhao
Keyword(s):  
Differential Equation ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Euler Equations ◽  
Direct Method ◽  
Special Solution ◽  
Linear Differential Equations ◽  
Linear Differential ◽  
Euler Solution ◽  
A Direct Method

For Euler Equations L(y）=∑aixiy(i) =f(x)are Given Special Solution of a Direct Method, and a Special Coefficient Linear Differential Equations L(y)=∑aiy(i)=∑bjejx into Ordinary Differential Equations Euler.

On the Camassa–Holm equation and a direct method of solution. III. N -soliton solutions

2005 ◽  
Vol 461 (2064) ◽  
pp. 3893-3911 ◽  
Author(s):  
Allen Parker
Keyword(s):  
Soliton Solution ◽  
Direct Method ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Bilinear Transformation ◽  
Standard Representation ◽  
Method Of Solution ◽  
A Direct Method ◽  
Multisoliton Solutions ◽  
Extra Parameter

In the concluding study (designated III), we modify the direct bilinear transformation method for solving the Camassa–Holm (CH) equation that was set down in part II of this work. We demonstrate its efficacy for finding analytic multisoliton solutions of the equation and give explicit expressions for the first few solitons. It is shown that, at each order N , the N -soliton has a non-standard representation that is characterized by an ‘extra’ parameter. The stucture of this parameter is investigated and a procedure for constructing the general N -soliton solution of the CH equation is presented.

On the Camassa–Holm equation and a direct method of solution. II. Soliton solutions

2005 ◽  
Vol 461 (2063) ◽  
pp. 3611-3632 ◽  
Author(s):  
Allen Parker
Keyword(s):  
Direct Method ◽  
Soliton Solutions ◽  
Transformation Method ◽  
Additional Parameter ◽  
Bilinear Transformation ◽  
Structure And Dynamics ◽  
Method Of Solution ◽  
A Direct Method ◽  
Multisoliton Solutions ◽  
Key Parameter

Previous attempts to find explicit analytic multisoliton solutions of the general Camassa–Holm (CH) equation have met with limited success. This study (which falls into two parts, designated II and III) extends the results of the prior work (I) in which a bilinear form of the CH equation was constructed and then solved for the solitary-wave solutions. It is shown that Hirota's bilinear transformation method can be used to derive exact multisoliton solutions of the equation in a systematic way. Here, analytic two-soliton solutions are obtained explicitly and their structure and dynamics are investigated in the different parameter regimes, including the limiting ‘two-peakon’ form. The solutions possess a non-standard representation that is characterized by an additional parameter, and the structure of this key parameter is examined. These results pave the way for constructing the hallmark N -soliton solutions of the CH equation in part III.

scholarly journals Discontinuous fluid motion past circular and elliptic cylinders

1923 ◽  
Vol 102 (718) ◽  
pp. 542-553 ◽  
Keyword(s):  
Numerical Calculation ◽  
Direct Method ◽  
Fluid Motion ◽  
Practical Applications ◽  
Method Of Solution ◽  
Christoffel Transformation ◽  
A Direct Method ◽  
Streaming Motion ◽  
The Way

1. The problem of discontinuous fluid motion past a curved barrier has become one of the classical problems of mechanics. For plane barriers the Schwartz-Christoffel transformation offers a direct method of solution; in the case of curved barriers no direct method of solution has been found. “A quicker start . . . can be made . . . by the simpler process of writing down a likely expression . . . and then investigating the streaming motion implied and the shape of the boundary.” In the present paper the procedure is essentially in pursuance of Greenhill’s advice, with the important modification that an attempt is made to solve problems that have a bearing on practical applications. Although many types of barriers have been suggested by various writers, very little in the way of actual numerical calculation has been done. Apparently, the only case worked out in detail is that by Brillouin, and this case is somewhat artificial and of little use. Such barriers as the circular and elliptic cylinders—important for the aerodynamics of struts—have not been attempted at all, while application to a barrier like a modern aeroplane wing seems very remote indeed.

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