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Differential Quadrature Method: A Robust Technique to Solve Differential Equations
Published in Mangey Ram, S. B. Singh, Mathematics Applied to Engineering and Management, 2020
Partial differential equations (PDEs), along with a set of initial or boundary conditions, play an important role in finding the solution of engineering problems. Most of the differential equations arise as a result of mathematical modeling of the various phenomenon of science. The solution of the PDEs is always required to obtain the values of the related parameters in the domain of interest. For example, the well-known Fisher’s equation is used to study the mutation rate of genes to analyze the spread of creature or plant population and to analyze the advancement of neutron population in an atomic reactor. Similarly, the solution of Burgers’ equation plays a significant role in the investigation of turbulence revealed by the association of the convection and dispersion phenomenon. Because it is not always possible to find a closed form solution of the equation using an analytical approach, there exists a need for a numerical approach.
Numerical Solution of Fisher's Equation by using Fourth-Order Collocation Scheme Based on Modified Cubic B-Splines
Published in Ziya Uddin, Mukesh Kumar Awasthi, Rishi Asthana, Mangey Ram, Computing and Simulation for Engineers, 2022
Brajesh Kumar Singh, Mukesh Gupta
In the field of science and engineering, Fisher’s equation (14.1) has a vital role in describing different phenomena such as in study of nuclear reactor theory [2], flame propagation in a medium [3], Brownian motion [4], autocatalytic chemical reactions [5], tissue engineering [6], neurophysiology [7], and combustion [8].
Feedback control for a class of semi-linear parabolic distributed parameter systems with mixed time delays
Published in International Journal of Systems Science, 2020
Huihui Ji, He Zhang, Baotong Cui
It is well known that the famous Fisher equation plays an important role in many fields, because it is frequently used to model heat and reaction-diffusion problems applied to heat and mass transfer, combustion theory, mathematical biology, astrophysics, chemistry, genetics, ecology, bacterial growth problems as well as development and growth of solid tumours. In this section, an illustrative example is given to demonstrate the effectiveness of the proposed design method. Motivated by Wang and Wu (2014), we consider the feedback control problem of a Fisher equation of the following form: where denotes the state variable, is a known parameter. stands for the initial value of the system. is the control distribution function and is the manipulated inputs. The kinetic function models also an autocatalytic chain reaction in combustion theory.