scholarly journals Similarity Solutions to Nonlinear Diffusion/Harry Dym Fractional Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
Chao Yue ◽  
Guijuan Liu ◽  
Kun Li ◽  
Hanhui Dong
Keyword(s):  
Differential Equations ◽  
Nonlinear Model ◽  
Numerical Solutions ◽  
Similarity Transformation ◽  
Fractional Diffusion ◽  
Travelling Wave ◽  
Similarity Solutions ◽  
Fractional Equations ◽  
Dym Equation ◽  
Harry Dym Equation

By using scalar similarity transformation, nonlinear model of time-fractional diffusion/Harry Dym equation is transformed to corresponding ordinary fractional differential equations, from which a travelling-wave similarity solution of time-fractional Harry Dym equation is presented. Furthermore, numerical solutions of time-fractional diffusion equation are discussed. Again, through another similarity transformation, nonlinear model of space-fractional diffusion/Harry Dym equation is turned into corresponding ordinary differential equations, whose two similarity solutions are also worked out.

scholarly journals Serious Solutions for Unsteady Axisymmetric Flow over a Rotating Stretchable Disk with Deceleration

Symmetry ◽  
2020 ◽  
Vol 12 (1) ◽  
pp. 96 ◽  
Author(s):  
Sadiq
Keyword(s):  
Differential Equations ◽  
Numerical Solutions ◽  
Similarity Transformation ◽  
Nonlinear Partial Differential Equations ◽  
Axisymmetric Flow ◽  
Analysis Method ◽  
Highly Nonlinear ◽  
Homotopy Analysis ◽  
Rotating Stretchable Disk ◽  
Comparison Table

In this article, the author has examined the unsteady flow over a rotating stretchable disk with deceleration. The highly nonlinear partial differential equations of viscous fluid are simplified by existing similarity transformation. Reduced nonlinear ordinary differential equations are solved by homotopy analysis method (HAM). The convergence of HAM solutions is also obtained. A comparison table between analytical solutions and numerical solutions is also presented. Finally, the results for useful parameters, i.e., disk stretching parameters and unsteadiness parameters, are found.

scholarly journals Magnetized Flow of Cu + Al2O3 + H2O Hybrid Nanofluid in Porous Medium: Analysis of Duality and Stability

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1513
Author(s):  
Liaquat Ali Lund ◽  
Zurni Omar ◽  
Sumera Dero ◽  
Ilyas Khan ◽  
Dumitru Baleanu ◽  
...  
Keyword(s):  
Porous Medium ◽  
Differential Equations ◽  
Similarity Transformation ◽  
Fluid Model ◽  
Flow Characteristics ◽  
Stable Solution ◽  
Similarity Solutions ◽  
Medium Effect ◽  
Shrinking Sheet ◽  
Hybrid Nanofluid

In this analysis, we aim to examine the heat transfer and flow characteristics of a copper-aluminum/water hybrid nanofluid in the presence of viscous dissipation, magnetohydrodynamic (MHD), and porous medium effect over the shrinking sheet. The governing equations of the fluid model have been acquired by employment of the model of Tiwari and Das, with additional properties of the hybrid nanofluid. The system of partial differential equations (PDEs) has been converted into ordinary differential equations (ODEs) by adopting the exponential similarity transformation. Similarity transformation is an essential class of phenomenon where the symmetry of the scale helps to reduce the number of independent variables. Note that ODE solutions demonstrate the PDEs symmetrical behavior for the velocity and temperature profiles. With BVP4C solver in the MATLAB program, the system of resulting equations has been solved. We have compared the present results with the published results and found in excellent agreements. The findings of the analysis are also displayed and discussed in depth graphically and numerically. It is discovered that two solutions occur in definite ranges of suction and magnetic parameters. Dual (no) similarity solutions can be found in the range of Sc≤S and Mc≤M (Sc>S and Mc>M). By performing stability analysis, the smallest values of eigenvalue are obtained, suggesting that a stable solution is the first one. Furthermore, the graph of the smallest eigenvalue shows symmetrical behavior. By enhancing the Eckert number values the temperature of the fluid is raised.

scholarly journals Numerical Solutions with Linearization Techniques of the Fractional Harry Dym Equation

2019 ◽  
Vol 4 (1) ◽  
pp. 35-42 ◽  
Author(s):  
Asıf Yokuş ◽  
Sema Gülbahar
Keyword(s):  
Finite Difference ◽  
Finite Difference Method ◽  
Difference Method ◽  
Numerical Solutions ◽  
Von Neumann ◽  
Non Linear ◽  
The Finite Difference Method ◽  
Dym Equation ◽  
Harry Dym Equation ◽  
Linearization Techniques

AbstractIn this study, numerical solutions of the fractional Harry Dym equation are investigated. Linearization techniques are utilized for non-linear terms existing in the fractional Harry Dym equation. The error norms L2 and L∞ are computed. Stability of the finite difference method is studied with the aid of Von Neumann stabity analysis.

scholarly journals An Efficient Approach for Fractional Harry Dym Equation by Using Sumudu Transform

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Devendra Kumar ◽  
Jagdev Singh ◽  
A. Kılıçman
Keyword(s):  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Numerical Solutions ◽  
Efficient Approach ◽  
Homotopy Perturbation ◽  
He’S Polynomials ◽  
Adomian Decomposition ◽  
Sumudu Transform ◽  
Dym Equation ◽  
Harry Dym Equation

An efficient approach based on homotopy perturbation method by using sumudu transform is proposed to solve nonlinear fractional Harry Dym equation. This method is called homotopy perturbation sumudu transform (HPSTM). Furthermore, the same problem is solved by Adomian decomposition method (ADM). The results obtained by the two methods are in agreement, and, hence, this technique may be considered an alternative and efficient method for finding approximate solutions of both linear and nonlinear fractional differential equations. The HPSTM is a combined form of sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The numerical solutions obtained by the HPSTM show that the approach is easy to implement and computationally very attractive.

Non-Newtonian Magneto-Hydrodynamic Fluid with Radiation by Stretching Cylinder

2020 ◽  
Vol 9 (2) ◽  
pp. 106-113
Author(s):  
Gamal M. Abdel-Rahman ◽  
Amal M. Al-Hanaya
Keyword(s):  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Similarity Transformation ◽  
Physical Parameters ◽  
Temperature Profiles ◽  
Stretching Cylinder ◽  
Linear Ordinary Differential Equations ◽  
Non Linear ◽  
Energy Equations

The aim of the present paper is to study the numerical solutions of the steady magnetohydrodynamic and heat generation effects on anon-Newtonian fluid with radiation through a porous medium by a stretching cylinder. The governing continuity, momentum, and energy equations are converted into a system of non-linear ordinary differential equations by means of similarity transformation. The resulting system of coupled non-linear ordinary differential equations is solved numerically. Numerical results were presented for velocity and temperature profiles for different parameters of the problem are studied graphically. Finally, the effects of related physical parameters on skin friction and the rate of heat transfer are also studied.

scholarly journals Some Similarity Solutions and Numerical Solutions to the Time-Fractional Burgers System

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 112 ◽  
Author(s):  
Xiangzhi Zhang ◽  
Yufeng Zhang
Keyword(s):  
Numerical Analysis ◽  
Differential Equations ◽  
Ordinary Differential Equations ◽  
Numerical Solutions ◽  
Similarity Solutions ◽  
Analysis Method ◽  
Liouville Type ◽  
Integer Order ◽  
Time Fractional Derivative ◽  
Burgers System

In the paper, we discuss some similarity solutions of the time-fractional Burgers system (TFBS). Firstly, with the help of the Lie-point symmetry and the corresponding invariant variables, we transform the TFBS to a fractional ordinary differential system (FODS) under the case where the time-fractional derivative is the Riemann–Liouville type. The FODS can be approximated by some integer-order ordinary differential equations; here, we present three such integer-order ordinary differential equations (called IODE-1, IODE-2, and IODE-3, respectively). For IODE-1, we obtain its similarity solutions and numerical solutions, which approximate the similarity solutions and the numerical solutions of the TFBS. Secondly, we apply the numerical analysis method to obtain the numerical solutions of IODE-2 and IODE-3.

The Effect of Suction on the Swirling Flow of Non-Newtonian Fluid

2014 ◽  
Vol 501-504 ◽  
pp. 2081-2084
Author(s):  
Chun Ying Ming ◽  
Lian Cun Zheng ◽  
Xin Xin Zhang
Keyword(s):  
Differential Equations ◽  
Power Law ◽  
Shooting Method ◽  
Numerical Solutions ◽  
Swirling Flow ◽  
Similarity Transformation ◽  
Rotating Disk ◽  
Shear Thinning Fluids ◽  
Suction Or Injection ◽  
Power Law Index

The flow of an incompressible viscous power-law fluid over an infinite rotating disk with uniform suction or injection is studied. The governing differential equations, which are partial and coupled, are simplified to a set of ordinary differential equations by generalized Karman similarity transformation. Numerical solutions of the non-linear two point boundary value problem are obtained by multi-shooting method. The effects of the power-law index and the porous parameter on the velocity fields are discussed for shear thinning fluids.

scholarly journals The space-fractional diffusion-advection equation: Analytical solutions and critical assessment of numerical solutions

2014 ◽  
Vol 17 (1) ◽  
Author(s):  
Robin Stern ◽  
Frederic Effenberger ◽  
Horst Fichtner ◽  
Tobias Schäfer
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Analytical Solutions ◽  
Numerical Solutions ◽  
Fractional Diffusion ◽  
Critical Assessment ◽  
Test Problems ◽  
Advection Equation ◽  
Standard Methods ◽  
Diffusion Problems

AbstractThe present work provides a critical assessment of numerical solutions of the space-fractional diffusion-advection equation, which is of high significance for applications in various natural sciences. In view of the fact that, in contrast to the case of normal (Gaussian) diffusion, no standard methods and corresponding numerical codes for anomalous diffusion problems have been established yet, it is of importance to critically assess the accuracy and practicability of existing approaches. Three numerical methods, namely a finite-difference method, the so-called matrix transfer technique, and a Monte-Carlo method based on the solution of stochastic differential equations, are analyzed and compared by applying them to three selected test problems for which analytical or semi-analytical solutions were known or are newly derived. The differences in accuracy and practicability are critically discussed with the result that the use of stochastic differential equations appears to be advantageous.

scholarly journals Computational Challenge of Fractional Differential Equations and the Potential Solutions: A Survey

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Chunye Gong ◽  
Weimin Bao ◽  
Guojian Tang ◽  
Yuewen Jiang ◽  
Jie Liu
Keyword(s):  
Parallel Computing ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Computer Science ◽  
Fractional Differential Equations ◽  
Numerical Solutions ◽  
Computational Cost ◽  
Variable Order ◽  
Fractional Equations ◽  
Fractional Differential

We present a survey of fractional differential equations and in particular of the computational cost for their numerical solutions from the view of computer science. The computational complexities of time fractional, space fractional, and space-time fractional equations areO(N2M),O(NM2), andO(NM(M+N)) compared withO(MN) for the classical partial differential equations with finite difference methods, whereM,Nare the number of space grid points and time steps. The potential solutions for this challenge include, but are not limited to, parallel computing, memory access optimization (fractional precomputing operator), short memory principle, fast Fourier transform (FFT) based solutions, alternating direction implicit method, multigrid method, and preconditioner technology. The relationships of these solutions for both space fractional derivative and time fractional derivative are discussed. The authors pointed out that the technologies of parallel computing should be regarded as a basic method to overcome this challenge, and some attention should be paid to the fractional killer applications, high performance iteration methods, high order schemes, and Monte Carlo methods. Since the computation of fractional equations with high dimension and variable order is even heavier, the researchers from the area of mathematics and computer science have opportunity to invent cornerstones in the area of fractional calculus.

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