Numerical Method for Simulation of Physical Processes Represented by Stiff and Nonstiff Fractional-Order Differential Equations, and Differential-Algebraic Equations

2018 ◽  
pp. 135-190
Author(s):  
Seshu Kumar Damarla
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Order ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Physical Processes ◽  
Fractional Order Differential Equations

scholarly journals Application of Said Ball Curve for Solving Fractional Differential-Algebraic Equations

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1926
Author(s):  
Fateme Ghomanjani ◽  
Samad Noeiaghdam
Keyword(s):  
Approximate Solution ◽  
Differential Equations ◽  
Fractional Order ◽  
Convergence Theorem ◽  
Differential Algebraic Equations ◽  
Bézier Curve ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
Curve Method ◽  
Fractional Differential

The aim of this paper is to apply the Said Ball curve (SBC) to find the approximate solution of fractional differential-algebraic equations (FDAEs). This method can be applied to solve various types of fractional order differential equations. Convergence theorem of the method is proved. Some examples are presented to show the efficiency and accuracy of the method. Based on the obtained results, the SBC is more accurate than the Bezier curve method.

scholarly journals A Numerical Method for Delayed Fractional-Order Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Zhen Wang
Keyword(s):  
Differential Equations ◽  
Numerical Method ◽  
Fractional Order ◽  
Exact Analytical Solution ◽  
Interpolation Method ◽  
Linear Interpolation ◽  
Positive Real ◽  
Time Varying Delay ◽  
Fractional Order Differential Equations ◽  
Varying Delay

A numerical method for nonlinear fractional-order differential equations with constant or time-varying delay is devised. The order here is an arbitrary positive real number, and the differential operator is with the Caputo definition. The general Adams-Bashforth-Moulton method combined with the linear interpolation method is employed to approximate the delayed fractional-order differential equations. Meanwhile, the detailed error analysis for this algorithm is given. In order to compare with the exact analytical solution, a numerical example is provided to illustrate the effectiveness of the proposed method.

Simple Recipe for Accurate Solution of Fractional Order Equations

2012 ◽  
Vol 8 (3) ◽  
Author(s):  
Sambit Das ◽  
Anindya Chatterjee
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Initial Conditions ◽  
Special Functions ◽  
Differential Algebraic Equations ◽  
Accurate Solution ◽  
Algebraic Equations ◽  
Matlab Program ◽  
Fractional Differential

Fractional order integrodifferential equations cannot be directly solved like ordinary differential equations. Numerical methods for such equations have additional algorithmic complexities. We present a particularly simple recipe for solving such equations using a Galerkin scheme developed in prior work. In particular, matrices needed for that method have here been precisely evaluated in closed form using special functions, and a small Matlab program is provided for the same. For equations where the highest order of the derivative is fractional, differential algebraic equations arise; however, it is demonstrated that there is a simple regularization scheme that works for these systems, such that accurate solutions can be easily obtained using standard solvers for stiff differential equations. Finally, the role of nonzero initial conditions is discussed in the context of the present approximation method.

Sixth-Kind Chebyshev Spectral Approach for Solving Fractional Differential Equations

2019 ◽  
Vol 20 (2) ◽  
pp. 191-203 ◽  
Author(s):  
W. M. Abd-Elhameed ◽  
Y. H. Youssri
Keyword(s):  
Differential Equations ◽  
Fractional Order ◽  
Chebyshev Polynomials ◽  
Symmetric Functions ◽  
Numerical Algorithms ◽  
Initial Boundary ◽  
Algebraic Equations ◽  
Fractional Order Differential Equations ◽  
New Type ◽  
Connection Formulae

AbstractThe basic aim of this paper is to develop new numerical algorithms for solving some linear and nonlinear fractional-order differential equations. We have developed a new type of Chebyshev polynomials, namely, Chebyshev polynomials of sixth kind. This type of polynomials is a special class of symmetric orthogonal polynomials, involving four parameters that were constructed with the aid of the extended Sturm–Liouville theorem for symmetric functions. The proposed algorithms are basically built on reducing the fractional-order differential equations with their initial/boundary conditions to systems of algebraic equations which can be efficiently solved. The new proposed algorithms are supported by a detailed study of the convergence and error analysis of the sixth-kind Chebyshev expansion. New connection formulae between Chebyshev polynomials of the second and sixth kinds were established for this study. Some examples were displayed to illustrate the efficiency of the proposed algorithms compared to other methods in literature. The proposed algorithms have provided accurate results, even using few terms of the proposed expansion.

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