A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

2012 ◽  
Vol 7 (2) ◽  
Author(s):  
Om P. Agrawal ◽  
M. Mehedi Hasan ◽  
X. W. Tangpong
Keyword(s):  
Analytical Solutions ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimization Problems ◽  
Practical Interest ◽  
Variational Calculus ◽  
Functional Minimization ◽  
Orthogonal Functions ◽  
Order Problem ◽  
Parametric Problem

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in the future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such properties as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered, and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

A Numerical Scheme for a Class of Parametric Problem of Fractional Variational Calculus

2011 ◽  
Author(s):  
Om P. Agrawal ◽  
Md. Mehedi Hasan ◽  
X. W. Tangpong
Keyword(s):  
Analytical Solutions ◽  
Numerical Scheme ◽  
Fractional Derivatives ◽  
Optimization Problems ◽  
Practical Interest ◽  
Variational Calculus ◽  
Functional Minimization ◽  
Orthogonal Functions ◽  
Order Problem ◽  
Parametric Problem

Fractional derivatives (FDs) or derivatives of arbitrary order have been used in many applications, and it is envisioned that in future they will appear in many functional minimization problems of practical interest. Since fractional derivatives have such property as being non-local, it can be extremely challenging to find analytical solutions for fractional parametric optimization problems, and in many cases, analytical solutions may not exist. Therefore, it is of great importance to develop numerical methods for such problems. This paper presents a numerical scheme for a linear functional minimization problem that involves FD terms. The FD is defined in terms of the Riemann-Liouville definition; however, the scheme will also apply to Caputo derivatives, as well as other definitions of fractional derivatives. In this scheme, the spatial domain is discretized into several subdomains and 2-node one-dimensional linear elements are adopted to approximate the solution and its fractional derivative at point within the domain. The fractional optimization problem is converted to an eigenvalue problem, the solution of which leads to fractional orthogonal functions. Convergence study of the number of elements and error analysis of the results ensure that the algorithm yields stable results. Various fractional orders of derivative are considered and as the order approaches the integer value of 1, the solution recovers the analytical result for the corresponding integer order problem.

Aspects on viscoelasticity modeling of HDPE using fractional derivatives: Interpolation procedures and efficient numerical scheme

2021 ◽  
pp. 1-16
Author(s):  
T.C. da Costa-Haveroth ◽  
G.A. Haveroth ◽  
A. Kühl ◽  
J.L. Boldrini ◽  
M.L. Bittencourt ◽  
...  
Keyword(s):  
Numerical Scheme ◽  
Fractional Derivatives

Optimum Oil Production Planning Using Infeasibility Driven Evolutionary Algorithm

2013 ◽  
Vol 21 (1) ◽  
pp. 65-82 ◽  
Author(s):  
Hemant Kumar Singh ◽  
Tapabrata Ray ◽  
Ruhul Sarker
Keyword(s):  
Evolutionary Algorithm ◽  
Production Planning ◽  
Oil Recovery ◽  
Optimization Problems ◽  
Nonlinear Function ◽  
Practical Interest ◽  
Oil Production ◽  
Economic Benefits ◽  
Financial Benefit ◽  
Single Objective

In this paper, we discuss a practical oil production planning optimization problem. For oil wells with insufficient reservoir pressure, gas is usually injected to artificially lift oil, a practice commonly referred to as enhanced oil recovery (EOR). The total gas that can be used for oil extraction is constrained by daily availability limits. The oil extracted from each well is known to be a nonlinear function of the gas injected into the well and varies between wells. The problem is to identify the optimal amount of gas that needs to be injected into each well to maximize the amount of oil extracted subject to the constraint on the total daily gas availability. The problem has long been of practical interest to all major oil exploration companies as it has the potential to derive large financial benefit. In this paper, an infeasibility driven evolutionary algorithm is used to solve a 56 well reservoir problem which demonstrates its efficiency in solving constrained optimization problems. Furthermore, a multi-objective formulation of the problem is posed and solved using a number of algorithms, which eliminates the need for solving the (single objective) problem on a regular basis. Lastly, a modified single objective formulation of the problem is also proposed, which aims to maximize the profit instead of the quantity of oil. It is shown that even with a lesser amount of oil extracted, more economic benefits can be achieved through the modified formulation.

scholarly journals New Analytical Solutions of Fractional Symmetric Regularized-Long-Wave Equation

2020 ◽  
Vol 66 (3 May-Jun) ◽  
pp. 297
Author(s):  
Mehmet Senol
Keyword(s):  
Wave Equation ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Algebraic Method ◽  
Symbolic Calculation ◽  
Solution Sets ◽  
Flow Models ◽  
Long Wave ◽  
Regularized Long Wave Equation ◽  
Fractional Differential

In this study, new extended direct algebraic method is successfully implemented to acquire new exact wave solution sets for symmetric regularized-long-wave (SRLW) equation which arise in long water flow models. By the help of Mathematica symbolic calculation package, the method produced a great number of analytical solutions. We also presented a few graphical illustrations for some surfaces. The fractional derivatives are considered in the conformable sense. All of the solutions were checked by substitution to ensure the reliability of the method. Obtained results confirm that the method is straightforward, powerful and effective method to attain exact solutions for nonlinear fractional differential equations. Therefore, the method is a good candidate to take part in the existing literature.

Applications of Creep Tests

1933 ◽  
Vol 1 (3) ◽  
pp. 87-97
Author(s):  
Gleason H. MacCullough
Keyword(s):  
Stress Distribution ◽  
Internal Pressure ◽  
Analytical Solutions ◽  
Practical Interest ◽  
Creep Data ◽  
Pure Bending ◽  
Creep Tests ◽  
Circular Shaft ◽  
Pipe Joint

Abstract Analytical solutions of problems which involve creep phenomena and which are of practical interest are at present very limited in number. This paper discusses four specific problems for which solutions have been presented: namely, the problem of the flanged and bolted pipe joint under creep conditions, and the three problems of stress distribution and creep in thick-walled cylinders under internal pressure, in a beam subjected to pure bending, and in a solid circular shaft under torsion. These solutions will illustrate the kind of creep data which the designer desires the experimenter to furnish.

scholarly journals LOCAL AND NONLOCAL DIFFERENTIAL OPERATORS: A COMPARATIVE STUDY OF HEAT AND MASS TRANSFERS IN MHD OLDROYD-B FLUID WITH RAMPED WALL TEMPERATURE

Fractals ◽  
2020 ◽  
Vol 28 (08) ◽  
pp. 2040033 ◽  
Author(s):  
MUHAMMAD BILAL RIAZ ◽  
ABDON ATANGANA ◽  
THABET ABDELJAWAD
Keyword(s):  
Wall Temperature ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Differential Operators ◽  
Classical Mechanics ◽  
Rate Of Change ◽  
Dynamical Process ◽  
New Class ◽  
Mass Transfers ◽  
And Inversion

Study of heat and mass transfers is carried out for MHD Oldroyd-B fluid (OBF) over an infinite vertical plate having time-dependent velocity and with ramped wall temperature and constant concentration. It is proven in many already published articles that the heat and mass transfers do not really or always follow the classical mechanics process that is known as memoryless process. Therefore, the model using classical differentiation based on the rate of change cannot really replicate such dynamical process very accurately, thus, a different concept of differentiation is needed to capture such process. Very recently, a new class of differential operators were introduced and have been recognized to be efficient in capturing processes following the power-law, the decay law and the crossover behaviors. For the study of heat and mass transfers, we applied the newly introduced differential operators to model such flow and compare the results with integer-order derivative. Laplace transform and inversion algorithms are used for all the cases to find analytical solutions and to predict the influences of different parameters. The obtained analytical solutions were plotted for different values of fractional orders [Formula: see text] and [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] on the velocity field. In comparison, Atangana–Baleanu (ABC) fractional derivatives are found to be the best to explain the memory effects than the classical, Caputo (C) and Caputo–Fabrizio (CF) fractional derivatives. Some calculated values for Nusselt number and Sherwood number are presented in tables. Moreover, from the present solutions, the already published results were found as limiting cases.

scholarly journals Local Fractional Series Expansion Method for Solving Wave and Diffusion Equations on Cantor Sets

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Ai-Min Yang ◽  
Xiao-Jun Yang ◽  
Zheng-Biao Li
Keyword(s):  
Differential Equations ◽  
Series Expansion ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Expansion Method ◽  
Cantor Sets ◽  
Diffusion Equations ◽  
And Diffusion ◽  
Series Expansion Method

We proposed a local fractional series expansion method to solve the wave and diffusion equations on Cantor sets. Some examples are given to illustrate the efficiency and accuracy of the proposed method to obtain analytical solutions to differential equations within the local fractional derivatives.

scholarly journals Time Dimensional Consistency Aware Analysis of Voltage Mode and Current Mode Active Fractional Circuits

2019 ◽  
Vol 13 (1) ◽  
pp. 81-93
Author(s):  
Rawid Banchuin ◽  
Roungsan Chaisrichaoren
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Analytical Solutions ◽  
Fractional Derivatives ◽  
Current Mode ◽  
Equation Approach ◽  
Time Dimension ◽  
Current Mode Circuits ◽  
Time Component ◽  
Fractional Differential

In this research, the analysis of the active fractional circuits has been performed by using the fractional differential equation approach. Both voltage and current mode circuits have been taken into account.  The fractional time component parameters have been included in the derivative terms within the fractional differential equations. This is because the consistency in time dimension between the fractional derivative and the conventional one which is also related to the physical measurability, is concerned. The fractional derivatives have been interpreted in Caputo sense. The resulting analytical solutions of the time dimensional consistency aware fractional differential equations have been determined. We have found that the dimensional consistency between both sides of the equations of the solutions which cannot be achieved in the previous works, can be obtained. By applying different source terms to the obtained analytical solutions, the response of both voltage and current mode circuits have been determined and the behaviours of the circuits have been analysed. The fractional time constant and pole locations in the F-plane of these circuits have been determined. Their dynamic behaviours, stabilities have been analysed. Moreover, the discussion on circuit realizations with fractional capacitor has also been made.

Sign in / Sign up

Export Citation Format

Share Document