scholarly journals AN OPTIMAL HOMOTOPY ANALYSIS METHOD BASED ON PARTICLE SWARM OPTIMIZATION: APPLICATION TO FRACTIONAL-ORDER DIFFERENTIAL EQUATION

2016 ◽  
Vol 6 (1) ◽  
pp. 103-118 ◽  
Author(s):  
Li-Guo Yuan ◽  
◽  
Zeeshan Alam ◽  
Keyword(s):  
Differential Equation ◽  
Particle Swarm Optimization ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Order Differential Equation ◽  
Analysis Method ◽  
Fractional Order Differential Equation ◽  
Optimal Homotopy Analysis Method ◽  
Swarm Optimization ◽  
Homotopy Analysis

Homotopy analysis method for solving Abel differential equation of fractional order

Open Physics ◽  
2013 ◽  
Vol 11 (10) ◽  
Author(s):  
Hossein Jafari ◽  
Khosro Sayevand ◽  
Haleh Tajadodi ◽  
Dumitru Baleanu
Keyword(s):  
Differential Equation ◽  
Fractional Order ◽  
Homotopy Analysis Method ◽  
Numerical Results ◽  
Analysis Method ◽  
Homotopy Analysis ◽  
Abel Differential Equation

AbstractIn this study, the homotopy analysis method is used for solving the Abel differential equation with fractional order within the Caputo sense. Stabilityand convergence of the proposed approach is investigated. The numerical results demonstrate that the homotopy analysis method is accurate and readily implemented.

scholarly journals Remark On Optimal Homotopy Method: Application Towards Nano-Fluid Flow Narrating Differential Equations

2019 ◽  
Vol 1 (1) ◽  
Author(s):  
Khalil Ur Rehman
Keyword(s):  
Differential Equations ◽  
Short Communication ◽  
Order Differential Equation ◽  
Nano Particle ◽  
Analysis Method ◽  
Fractional Order Differential Equation ◽  
Nanofluid Flow ◽  
Optimal Homotopy Analysis Method ◽  
Nano Fluid ◽  
Homotopy Analysis

The short communication is devoted to validate the reliability and convergence of Optimal Homotopy Analysis Method (O-HAM). Owing the importance of present validation of O-HAM one can implement this method towards nanofluid flow narrating differential equations at larger scale for better description. To be more specific, the fractional order differential equation due to vertically moving non-spherical nano particle in a purely viscous liquid and an advection PDE is take into account. The corresponding homotopy for both cases are constructed and solutions are proposed by means of O-HAM. The obtained values are compared with numerical benchmarks. We observed an excellent match which confirms the O-HAM conjecture. Therefore, it can be directed that the utilization of O-HAM towards nanofluid flow regime may provide relief against some non-attempted problems.

Optimal Homotopic Exploration of features of Cattaneo-Christov model in Second Grade Nanofluid flow via Darcy-Forchheimer medium subject to Viscous Dissipation and Thermal Radiation

2021 ◽  
Vol 24 ◽  
Author(s):  
Ghulam Rasool ◽  
Anum Shafiq ◽  
Yu-Ming Chu ◽  
Muhammad Shoaib Bhutta ◽  
Amjad Ali
Keyword(s):  
Temperature Distribution ◽  
Differential Equations ◽  
Thermal Radiation ◽  
Skin Friction ◽  
Homotopy Analysis Method ◽  
Stream Velocity ◽  
Analysis Method ◽  
Nanofluid Flow ◽  
Optimal Homotopy Analysis Method ◽  
Homotopy Analysis

Introduction: In this article Optimal Homotopy analysis method (oHAM) is used for exploration of the features of Cattaneo-Christov model in viscous and chemically reactive nanofluid flow through a porous medium with stretching velocity at the solid/sheet surface and free stream velocity at the free surface. Methods: The two important aspects, Brownian motion and Thermophoresis are considered. Thermal radiation is also included in present model. Based on the heat and mass flux, the Cattaneo-Christov model is implemented on the Temperature and Concentration distributions. The governing Partial Differential Equations (PDEs) are converted into Ordinary Differential Equations (ODEs) using similarity transformations. The results are achieved using the optimal homotopy analysis method (oHAM). The optimal convergence and residual errors have been calculated to preserve the validity of the model. Results: The results are plotted graphically to see the variations in three main profiles i.e. momentum, temperature and concentration profile. Conclusion: The outcomes indicate that skin friction enhances due to implementation of Darcy medium. It is also noted that the relaxation time parameter results in enhancement of the temperature distribution. Thermal radiation enhances the temperature distribution and so is the case with skin friction.

scholarly journals Fractional-Derivative Approximation of Relaxation in Complex Systems

Complexity ◽  
2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Kin M. Li ◽  
Mihir Sen ◽  
Arturo Pacheco-Vega
Keyword(s):  
Differential Equation ◽  
Experimental Data ◽  
System Identification ◽  
Complex Systems ◽  
Fractional Derivative ◽  
Fractional Order ◽  
Order Differential Equation ◽  
Initial Conditions ◽  
Time Dependent ◽  
Fractional Order Differential Equation

In this paper, we present a system identification (SI) procedure that enables building linear time-dependent fractional-order differential equation (FDE) models able to accurately describe time-dependent behavior of complex systems. The parameters in the models are the order of the equation, the coefficients in it, and, when necessary, the initial conditions. The Caputo definition of the fractional derivative, and the Mittag-Leffler function, is used to obtain the corresponding solutions. Since the set of parameters for the model and its initial conditions are nonunique, and there are small but significant differences in the predictions from the possible models thus obtained, the SI operation is carried out via global regression of an error-cost function by a simulated annealing optimization algorithm. The SI approach is assessed by considering previously published experimental data from a shell-and-tube heat exchanger and a recently constructed multiroom building test bed. The results show that the proposed model is reliable within the interpolation domain but cannot be used with confidence for predictions outside this region. However, the proposed system identification methodology is robust and can be used to derive accurate and compact models from experimental data. In addition, given a functional form of a fractional-order differential equation model, as new data become available, the SI technique can be used to expand the region of reliability of the resulting model.

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