Series solutions of non-linear Riccati differential equations with fractional order

2009 ◽  
Vol 40 (1) ◽  
pp. 1-9 ◽  
Author(s):  
Jie Cang ◽  
Yue Tan ◽  
Hang Xu ◽  
Shi-Jun Liao
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Series Solutions ◽  
Riccati Differential Equations ◽  
Non Linear

AN EFFECTIVE NUMERICAL METHOD FOR SOLVING THE NONLINEAR SINGULAR LANE-EMDEN TYPE EQUATIONS OF VARIOUS ORDERS

2016 ◽  
Vol 79 (1) ◽  
Author(s):  
Kourosh Parand ◽  
Mehdi Delkhosh
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Method ◽  
Collocation Method ◽  
Fractional Order ◽  
Spectral Methods ◽  
Mathematical Physics ◽  
Infinite Domain ◽  
Orthogonal Functions ◽  
Non Linear

The Lane-Emden type equations are employed in the modeling of several phenomena in the areas of mathematical physics and astrophysics. These equations are categorized as non-linear singular ordinary differential equations on the semi-infinite domain. In this paper, the generalized fractional order of the Chebyshev orthogonal functions (GFCFs) of the first kind have been introduced as a new basis for Spectral methods, and also presented an effective numerical method based on the GFCFs and the collocation method for solving the nonlinear singular Lane-Emden type equations of various orders. Obtained results have compared with other results to verify the accuracy and efficiency of the presented method.

Third-Kind Chebyshev Wavelet Method for the Solution of Fractional Order Riccati Differential Equations

2019 ◽  
Vol 28 (14) ◽  
pp. 1950247 ◽  
Author(s):  
Sadiye Nergis Tural-Polat
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Numerical Solutions ◽  
Operational Matrix ◽  
Algebraic Equations ◽  
Wavelet Method ◽  
Riccati Differential Equations ◽  
The Third ◽  
Chebyshev Wavelet ◽  
Fractional Orders

In this paper, we derive the numerical solutions of the various fractional-order Riccati type differential equations using the third-kind Chebyshev wavelet operational matrix of fractional order integration (C3WOMFI) method. The operational matrix of fractional order integration method converts the fractional differential equations to a system of algebraic equations. The third-kind Chebyshev wavelet method provides sparse coefficient matrices, therefore the computational load involved for this method is not as severe and also the resulting method is faster. The numerical solutions agree with the exact solutions for non-fractional orders, and also the solutions for the fractional orders approach those of the integer orders as the fractional order coefficient [Formula: see text] approaches to 1.

Oblique stagnation flow of Jeffery fluid over a stretching convective surface

2015 ◽  
Vol 25 (3) ◽  
pp. 454-471 ◽  
Author(s):  
R Mehmood ◽  
Dr. Sohail Nadeem ◽  
Noreen Akbar
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Convergent Series ◽  
Series Solutions ◽  
Content Type ◽  
Analytical Technique ◽  
Linear Partial Differential Equations ◽  
Similarity Transformations ◽  
Non Linear ◽  
Linear Differential

Purpose – The present critical analysis has been performed to explore the steady stagnation point flow of Jeffery fluid toward a stretching surface, in the presence of convective boundary conditions. It is assumed that the fluid strikes the wall obliquely. The governing non-linear partial differential equations for the flow field are converted to ordinary differential equations by using suitable similarity transformations. Optimal homotopy analysis method (OHAM) is operated to deal the resulting ordinary differential equations. OHAM is found to be extremely effective analytical technique to obtain convergent series solutions of highly non-linear differential equations. Graphically, non-dimensional velocities and temperature profile are expressed. Numerical values of skin friction coefficients and heat flux are computed. The comparison of results from this paper with the previous existing literature authorizes the precise accuracy of the OHAM for the limited case. The paper aims to discuss these issues. Design/methodology/approach – The governing non-linear partial differential equations for the flow field are converted to ordinary differential equations by using suitable similarity transformations. OHAM is operated to deal the resulting ordinary differential equations. Findings – OHAM is found to be extremely effective analytical technique to obtain convergent series solutions of highly non-linear differential equations. Graphically, non-dimensional velocities and temperature profile are expressed. Numerical values of skin friction coefficients and heat flux are computed. Originality/value – The comparison of results from this paper with the previous existing literature authorizes the precise accuracy of the OHAM for the limited case.

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