Editorial

4 pages, 165 KiB

Open AccessEditorial

Special Issue on Mathematical Modeling Using Differential Equations and Network Theory

by Ioannis Dassios

Appl. Sci. 2020, 10(5), 1895; https://0-doi-org.brum.beds.ac.uk/10.3390/app10051895 - 10 Mar 2020

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This special issue collects the latest results on differential/difference equations, the mathematics of networks, and their applications to engineering, and physical phenomena [...] Full article

(This article belongs to the Special Issue Mathematical Modeling using Differential Equations, and Network Theory)

Research

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17 pages, 572 KiB

Open AccessArticle

Fractional View Analysis of Acoustic Wave Equations, Using Fractional-Order Differential Equations

by Izaz Ali, Hassan Khan, Rasool Shah, Dumitru Baleanu, Poom Kumam and Muhammad Arif

Appl. Sci. 2020, 10(2), 610; https://0-doi-org.brum.beds.ac.uk/10.3390/app10020610 - 15 Jan 2020

Cited by 12 | Viewed by 2217

Abstract

In the present research work, a newly developed technique which is known as variational homotopy perturbation transform method is implemented to solve fractional-order acoustic wave equations. The basic idea behind the present research work is to extend the variational homotopy perturbation method to [...] Read more.

In the present research work, a newly developed technique which is known as variational homotopy perturbation transform method is implemented to solve fractional-order acoustic wave equations. The basic idea behind the present research work is to extend the variational homotopy perturbation method to variational homotopy perturbation transform method. The proposed scheme has confirmed, that it is an accurate and straightforward technique to solve fractional-order partial differential equations. The validity of the method is verified with the help of some illustrative examples. The obtained solutions have shown close contact with the exact solutions. Furthermore, the highest degree of accuracy has been achieved by the suggested method. In fact, the present method can be considered as one of the best analytical techniques compared to other analytical techniques to solve non-linear fractional partial differential equations. Full article

(This article belongs to the Special Issue Mathematical Modeling using Differential Equations, and Network Theory)

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20 pages, 1823 KiB

Open AccessArticle

Analytical Solutions of (2+Time Fractional Order) Dimensional Physical Models, Using Modified Decomposition Method

by Hassan Khan, Umar Farooq, Rasool Shah, Dumitru Baleanu, Poom Kumam and Muhammad Arif

Appl. Sci. 2020, 10(1), 122; https://0-doi-org.brum.beds.ac.uk/10.3390/app10010122 - 23 Dec 2019

Cited by 34 | Viewed by 3431

Abstract

In this article, a new analytical technique based on an innovative transformation is used to solve (2+time fractional-order) dimensional physical models. The proposed method is the hybrid methodology of Shehu transformation along with Adomian decomposition method. The series form solution is obtained by [...] Read more.

In this article, a new analytical technique based on an innovative transformation is used to solve (2+time fractional-order) dimensional physical models. The proposed method is the hybrid methodology of Shehu transformation along with Adomian decomposition method. The series form solution is obtained by using the suggested method which provides the desired rate of convergence. Some numerical examples are solved by using the proposed method. The solutions of the targeted problems are represented by graphs which have confirmed closed contact between the exact and obtained solutions of the problems. Based on the novelty and straightforward implementation of the method, it is considered to be one of the best analytical techniques to solve linear and non-linear fractional partial differential equations. Full article

(This article belongs to the Special Issue Mathematical Modeling using Differential Equations, and Network Theory)

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17 pages, 1841 KiB

Open AccessArticle

Multi-Switching Combination Synchronization of Three Fractional-Order Delayed Systems

by Bo Li, Yun Wang and Xiaobing Zhou

Appl. Sci. 2019, 9(20), 4348; https://0-doi-org.brum.beds.ac.uk/10.3390/app9204348 - 15 Oct 2019

Cited by 3 | Viewed by 1777

Abstract

Multi-switching combination synchronization of three fractional-order delayed systems is investigated. This is a generalization of previous multi-switching combination synchronization of fractional-order systems by introducing time-delays. Based on the stability theory of linear fractional-order systems with multiple time-delays, we propose appropriate controllers to obtain [...] Read more.

Multi-switching combination synchronization of three fractional-order delayed systems is investigated. This is a generalization of previous multi-switching combination synchronization of fractional-order systems by introducing time-delays. Based on the stability theory of linear fractional-order systems with multiple time-delays, we propose appropriate controllers to obtain multi-switching combination synchronization of three non-identical fractional-order delayed systems. In addition, the results of our numerical simulations show that they are in accordance with the theoretical analysis. Full article

(This article belongs to the Special Issue Mathematical Modeling using Differential Equations, and Network Theory)

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19 pages, 1525 KiB

Open AccessArticle

Parametrical Non-Complex Tests to Evaluate Partial Decentralized Linear-Output Feedback Control Stabilization Conditions from Their Centralized Stabilization Counterparts

by Manuel De la Sen and Asier Ibeas

Appl. Sci. 2019, 9(9), 1739; https://0-doi-org.brum.beds.ac.uk/10.3390/app9091739 - 26 Apr 2019

Cited by 7 | Viewed by 1837

Abstract

This paper formulates sufficiency-type linear-output feedback decentralized closed-loop stabilization conditions if the continuous-time linear dynamic system can be stabilized under linear output-feedback centralized stabilization. The provided tests are simple to evaluate, while they are based on the quantification of the sufficiently smallness of [...] Read more.

This paper formulates sufficiency-type linear-output feedback decentralized closed-loop stabilization conditions if the continuous-time linear dynamic system can be stabilized under linear output-feedback centralized stabilization. The provided tests are simple to evaluate, while they are based on the quantification of the sufficiently smallness of the parametrical error norms between the control, output, interconnection and open-loop system dynamics matrices and the corresponding control gains in the decentralized case related to the centralized counterpart. The tolerance amounts of the various parametrical matrix errors are described by the maximum allowed tolerance upper-bound of a small positive real parameter that upper-bounds the various parametrical error norms. Such a tolerance is quantified by considering the first or second powers of such a small parameter. The results are seen to be directly extendable to quantify the allowed parametrical errors that guarantee the closed-loop linear output-feedback stabilization of a current system related to its nominal counterpart. Furthermore, several simulated examples are also discussed. Full article

(This article belongs to the Special Issue Mathematical Modeling using Differential Equations, and Network Theory)

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23 pages, 10112 KiB

Open AccessArticle

Transient-Flow Modeling of Vertical Fractured Wells with Multiple Hydraulic Fractures in Stress-Sensitive Gas Reservoirs

by Ping Guo, Zhen Sun, Chao Peng, Hongfei Chen and Junjie Ren

Appl. Sci. 2019, 9(7), 1359; https://0-doi-org.brum.beds.ac.uk/10.3390/app9071359 - 31 Mar 2019

Cited by 3 | Viewed by 2935

Abstract

Massive hydraulic fracturing of vertical wells has been extensively employed in the development of low-permeability gas reservoirs. The existence of multiple hydraulic fractures along a vertical well makes the pressure profile around the vertical well complex. This paper studies the pressure dependence of [...] Read more.

Massive hydraulic fracturing of vertical wells has been extensively employed in the development of low-permeability gas reservoirs. The existence of multiple hydraulic fractures along a vertical well makes the pressure profile around the vertical well complex. This paper studies the pressure dependence of permeability to develop a seepage model of vertical fractured wells with multiple hydraulic fractures. Both transformed pseudo-pressure and perturbation techniques have been employed to linearize the proposed model. The superposition principle and a hybrid analytical-numerical method were used to obtain the bottom-hole pseudo-pressure solution. Type curves for pseudo-pressure are presented and identified. The effects of the relevant parameters (such as dimensionless permeability modulus, fracture conductivity coefficient, hydraulic-fracture length, angle between the two adjacent hydraulic fractures, the difference of the hydraulic-fracture lengths, and hydraulic-fracture number) on the type curve and the error caused by neglecting the stress sensitivity are discussed in detail. The proposed work can enrich the understanding of the influence of the stress sensitivity on the performance of a vertical fractured well with multiple hydraulic fractures and can be used to more accurately interpret and forecast the transient pressure. Full article

(This article belongs to the Special Issue Mathematical Modeling using Differential Equations, and Network Theory)

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16 pages, 433 KiB

Open AccessArticle

Policy-Compliant Maximum Network Flows

by Pieter Audenaert, Didier Colle and Mario Pickavet

Appl. Sci. 2019, 9(5), 863; https://0-doi-org.brum.beds.ac.uk/10.3390/app9050863 - 28 Feb 2019

Cited by 2 | Viewed by 1946

Abstract

Computer network administrators are often interested in the maximal bandwidth that can be achieved between two nodes in the network, or how many edges can fail before the network gets disconnected. Classic maximum flow algorithms that solve these problems are well-known. However, in [...] Read more.

Computer network administrators are often interested in the maximal bandwidth that can be achieved between two nodes in the network, or how many edges can fail before the network gets disconnected. Classic maximum flow algorithms that solve these problems are well-known. However, in practice, network policies are in effect, severely restricting the flow that can actually be set up. These policies are put into place to conform to service level agreements and optimize network throughput, and can have a large impact on the actual routing of the flows. In this work, we model the problem and define a series of progressively more complex conditions and algorithms that calculate increasingly tighter bounds on the policy-compliant maximum flow using regular expressions and finite state automata. To the best of our knowledge, this is the first time that specific conditions are deduced, which characterize how to calculate policy-compliant maximum flows using classic algorithms on an unmodified network. Full article

(This article belongs to the Special Issue Mathematical Modeling using Differential Equations, and Network Theory)

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12 pages, 6395 KiB

Open AccessArticle

The Fractional Form of the Tinkerbell Map Is Chaotic

by Adel Ouannas, Amina-Aicha Khennaoui, Samir Bendoukha, Thoai Phu Vo, Viet-Thanh Pham and Van Van Huynh

Appl. Sci. 2018, 8(12), 2640; https://0-doi-org.brum.beds.ac.uk/10.3390/app8122640 - 16 Dec 2018

Cited by 22 | Viewed by 5646

Abstract

This paper is concerned with a fractional Caputo-difference form of the well-known Tinkerbell chaotic map. The dynamics of the proposed map are investigated numerically through phase plots, bifurcation diagrams, and Lyapunov exponents considered from different perspectives. In addition, a stabilization controller is proposed, [...] Read more.

This paper is concerned with a fractional Caputo-difference form of the well-known Tinkerbell chaotic map. The dynamics of the proposed map are investigated numerically through phase plots, bifurcation diagrams, and Lyapunov exponents considered from different perspectives. In addition, a stabilization controller is proposed, and the asymptotic convergence of the states is established by means of the stability theory of linear fractional discrete systems. Numerical results are employed to confirm the analytical findings. Full article

(This article belongs to the Special Issue Mathematical Modeling using Differential Equations, and Network Theory)

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Review

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15 pages, 1720 KiB

Open AccessReview

Short Overview of Early Developments of the Hardy Cross Type Methods for Computation of Flow Distribution in Pipe Networks

by Dejan Brkić and Pavel Praks

Appl. Sci. 2019, 9(10), 2019; https://0-doi-org.brum.beds.ac.uk/10.3390/app9102019 - 16 May 2019

Cited by 19 | Viewed by 4973

Abstract

Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the field of pipe network calculation. Only electrical analogs of hydraulic networks were used before [...] Read more.

Hardy Cross originally proposed a method for analysis of flow in networks of conduits or conductors in 1936. His method was the first really useful engineering method in the field of pipe network calculation. Only electrical analogs of hydraulic networks were used before the Hardy Cross method. A problem with flow resistance versus electrical resistance makes these electrical analog methods obsolete. The method by Hardy Cross is taught extensively at faculties, and it remains an important tool for the analysis of looped pipe systems. Engineers today mostly use a modified Hardy Cross method that considers the whole looped network of pipes simultaneously (use of these methods without computers is practically impossible). A method from a Russian practice published during the 1930s, which is similar to the Hardy Cross method, is described, too. Some notes from the work of Hardy Cross are also presented. Finally, an improved version of the Hardy Cross method, which significantly reduces the number of iterations, is presented and discussed. We also tested multi-point iterative methods, which can be used as a substitution for the Newton–Raphson approach used by Hardy Cross, but in this case this approach did not reduce the number of iterations. Although many new models have been developed since the time of Hardy Cross, the main purpose of this paper is to illustrate the very beginning of modeling of gas and water pipe networks and ventilation systems. As a novelty, a new multi-point iterative solver is introduced and compared with the standard Newton–Raphson iterative method. Full article

(This article belongs to the Special Issue Mathematical Modeling using Differential Equations, and Network Theory)

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