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Advanced Numerical Methods for Differential Equations

A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Multidisciplinary Applications".

Deadline for manuscript submissions: closed (15 April 2022) | Viewed by 26152

Special Issue Editors


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Guest Editor
1. Mathematics, Anand International College of Engineering, Jaipur, Rajasthan 303012, India
2. Nonlinear Dynamics Research Center (NDRC), Ajman University, Al Jerf 1, Ajman, United Arab Emirates
Interests: special functions; fractional calculus; integral transform; control theory
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Engineering School (DEIM), University of Tuscia, Largo dell'Università, 01100 Viterbo, Italy
Interests: wavelets; fractals; fractional and stochastic equations; numerical and computational methods; mathematical physics; nonlinear systems; artificial intelligence
Special Issues, Collections and Topics in MDPI journals

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Guest Editor
Computer Science Department, University of Georgia, 412 Boyd GSRC, Athens, GA 30602-7404, USA
Interests: numerical analysis and scientific computing; parallel algorithms

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Guest Editor
College of Humanities and Sciences, Ajman University, Ajman, United Arab Emirates
Interests: numerical analysis; differential equations; fractional calculus; fluid mechanics; nonlinear dynamics

Special Issue Information

Dear Colleagues,

Differential equations, in general, have attracted more and more attention in mathematical, scientific, and engineering communities due to their wide real-life applications in mathematical modeling physical/engineering/biological systems, and many other areas.

In general, it is difficult to solve some kind of mathematical model due to the complexity. These models are governed by differential equations whose solutions make it easy to understand real-life problems and can be applied to engineering and science disciplines.  This Special Issue is mainly focused to address a wide range of computational methods ranging from efficient finite element and finite difference methods, adaptive methods, multi-scale methods, to spectral methods and kinetic Monte Carlo simulations. Computational challenges will be discussed, and new computational techniques will be introduced for various applications. Engineers, mathematicians, scientists, and researchers working on real-life mathematical problems will find this special issue useful. 

Prof. Dr. Praveen Agarwal
Prof. Dr. Carlo Cattani
Prof. Dr. Thiab Taha
Prof. Dr. Shaher Momani
Prof. Dr. Juan Luis García Guirao
Guest Editors

Manuscript Submission Information

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Keywords

  • Numerical methods for differential equations
  • Ordinary differential equations
  • Fractional differential equations
  • Partial differential equations
  • Initial value problems
  • Boundary value problems
  • Singular differential equations
  • Singularly perturbed problems
  • Delay differential equations
  • Algebraic differential equations
  • Finite Difference Methods
  • Finite Volume Methods
  • Finite Element Methods
  • Integration Factor Methods
  • Operator-Splitting Methods

Published Papers (11 papers)

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Research

10 pages, 272 KiB  
Article
Quantum Estimates for Different Type Intequalities through Generalized Convexity
by Ohud Bulayhan Almutairi
Entropy 2022, 24(5), 728; https://doi.org/10.3390/e24050728 - 20 May 2022
Cited by 4 | Viewed by 1479
Abstract
This article estimates several integral inequalities involving (hm)-convexity via the quantum calculus, through which Important integral inequalities including Simpson-like, midpoint-like, averaged midpoint-trapezoid-like and trapezoid-like are extended. We generalized some quantum integral inequalities for q-differentiable [...] Read more.
This article estimates several integral inequalities involving (hm)-convexity via the quantum calculus, through which Important integral inequalities including Simpson-like, midpoint-like, averaged midpoint-trapezoid-like and trapezoid-like are extended. We generalized some quantum integral inequalities for q-differentiable (hm)-convexity. Our results could serve as the refinement and the unification of some classical results existing in the literature by taking the limit q1. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
15 pages, 2729 KiB  
Article
Numerical Solutions of Variable Coefficient Higher-Order Partial Differential Equations Arising in Beam Models
by Abdul Ghafoor, Sirajul Haq, Manzoor Hussain, Thabet Abdeljawad and Manar A. Alqudah
Entropy 2022, 24(4), 567; https://doi.org/10.3390/e24040567 - 18 Apr 2022
Cited by 1 | Viewed by 2059
Abstract
In this work, an efficient and robust numerical scheme is proposed to solve the variable coefficients’ fourth-order partial differential equations (FOPDEs) that arise in Euler–Bernoulli beam models. When partial differential equations (PDEs) are of higher order and invoke variable coefficients, then the numerical [...] Read more.
In this work, an efficient and robust numerical scheme is proposed to solve the variable coefficients’ fourth-order partial differential equations (FOPDEs) that arise in Euler–Bernoulli beam models. When partial differential equations (PDEs) are of higher order and invoke variable coefficients, then the numerical solution is quite a tedious and challenging problem, which is our main concern in this paper. The current scheme is hybrid in nature in which the second-order finite difference is used for temporal discretization, while spatial derivatives and solutions are approximated via the Haar wavelet. Next, the integration and Haar matrices are used to convert partial differential equations (PDEs) to the system of linear equations, which can be handled easily. Besides this, we derive the theoretical result for stability via the Lax–Richtmyer criterion and verify it computationally. Moreover, we address the computational convergence rate, which is near order two. Several test problems are given to measure the accuracy of the suggested scheme. Computations validate that the present scheme works well for such problems. The calculated results are also compared with the earlier work and the exact solutions. The comparison shows that the outcomes are in good agreement with both the exact solutions and the available results in the literature. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
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21 pages, 750 KiB  
Article
Exact and Numerical Solution of the Fractional Sturm–Liouville Problem with Neumann Boundary Conditions
by Malgorzata Klimek, Mariusz Ciesielski and Tomasz Blaszczyk
Entropy 2022, 24(2), 143; https://doi.org/10.3390/e24020143 - 18 Jan 2022
Cited by 10 | Viewed by 2037
Abstract
In this paper, we study the fractional Sturm–Liouville problem with homogeneous Neumann boundary conditions. We transform the differential problem to an equivalent integral one on a suitable function space. Next, we discretize the integral fractional Sturm–Liouville problem and discuss the orthogonality of eigenvectors. [...] Read more.
In this paper, we study the fractional Sturm–Liouville problem with homogeneous Neumann boundary conditions. We transform the differential problem to an equivalent integral one on a suitable function space. Next, we discretize the integral fractional Sturm–Liouville problem and discuss the orthogonality of eigenvectors. Finally, we present the numerical results for the considered problem obtained by utilizing the midpoint rectangular rule. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
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20 pages, 2868 KiB  
Article
Series Representations for Uncertain Fractional IVPs in the Fuzzy Conformable Fractional Sense
by Malik Bataineh, Mohammad Alaroud, Shrideh Al-Omari and Praveen Agarwal
Entropy 2021, 23(12), 1646; https://doi.org/10.3390/e23121646 - 7 Dec 2021
Cited by 23 | Viewed by 2142
Abstract
Fuzzy differential equations provide a crucial tool for modeling numerous phenomena and uncertainties that potentially arise in various applications across physics, applied sciences and engineering. Reliable and effective analytical methods are necessary to obtain the required solutions, as it is very difficult to [...] Read more.
Fuzzy differential equations provide a crucial tool for modeling numerous phenomena and uncertainties that potentially arise in various applications across physics, applied sciences and engineering. Reliable and effective analytical methods are necessary to obtain the required solutions, as it is very difficult to obtain accurate solutions for certain fuzzy differential equations. In this paper, certain fuzzy approximate solutions are constructed and analyzed by means of a residual power series (RPS) technique involving some class of fuzzy fractional differential equations. The considered methodology for finding the fuzzy solutions relies on converting the target equations into two fractional crisp systems in terms of ρ-cut representations. The residual power series therefore gives solutions for the converted systems by combining fractional residual functions and fractional Taylor expansions to obtain values of the coefficients of the fractional power series. To validate the efficiency and the applicability of our proposed approach we derive solutions of the fuzzy fractional initial value problem by testing two attractive applications. The compatibility of the behavior of the solutions is determined via some graphical and numerical analysis of the proposed results. Moreover, the comparative results point out that the proposed method is more accurate compared to the other existing methods. Finally, the results attained in this article emphasize that the residual power series technique is easy, efficient, and fast for predicting solutions of the uncertain models arising in real physical phenomena. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
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22 pages, 380 KiB  
Article
Simple Equations Method and Non-Linear Differential Equations with Non-Polynomial Non-Linearity
by Nikolay K. Vitanov and Zlatinka I. Dimitrova
Entropy 2021, 23(12), 1624; https://doi.org/10.3390/e23121624 - 2 Dec 2021
Cited by 13 | Viewed by 2668
Abstract
We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. [...] Read more.
We discuss the application of the Simple Equations Method (SEsM) for obtaining exact solutions of non-linear differential equations to several cases of equations containing non-polynomial non-linearity. The main idea of the study is to use an appropriate transformation at Step (1.) of SEsM. This transformation has to convert the non-polynomial non- linearity to polynomial non-linearity. Then, an appropriate solution is constructed. This solution is a composite function of solutions of more simple equations. The application of the solution reduces the differential equation to a system of non-linear algebraic equations. We list 10 possible appropriate transformations. Two examples for the application of the methodology are presented. In the first example, we obtain kink and anti- kink solutions of the solved equation. The second example illustrates another point of the study. The point is as follows. In some cases, the simple equations used in SEsM do not have solutions expressed by elementary functions or by the frequently used special functions. In such cases, we can use a special function, which is the solution of an appropriate ordinary differential equation, containing polynomial non-linearity. Specific cases of the use of this function are presented in the second example. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
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17 pages, 340 KiB  
Article
New Parameterized Inequalities for η-Quasiconvex Functions via (p, q)-Calculus
by Humaira Kalsoom, Miguel Vivas-Cortez, Muhammad Idrees and Praveen Agarwal
Entropy 2021, 23(11), 1523; https://doi.org/10.3390/e23111523 - 16 Nov 2021
Cited by 1 | Viewed by 1983
Abstract
In this work, first, we consider novel parameterized identities for the left and right part of the (p,q)-analogue of Hermite–Hadamard inequality. Second, using these new parameterized identities, we give new parameterized (p,q)-trapezoid and [...] Read more.
In this work, first, we consider novel parameterized identities for the left and right part of the (p,q)-analogue of Hermite–Hadamard inequality. Second, using these new parameterized identities, we give new parameterized (p,q)-trapezoid and parameterized (p,q)-midpoint type integral inequalities via η-quasiconvex function. By changing values of parameter μ[0,1], some new special cases from the main results are obtained and some known results are recaptured as well. Finally, at the end, an application to special means is given as well. This new research has the potential to establish new boundaries in comparative literature and some well-known implications. From an application perspective, the proposed research on the η-quasiconvex function has interesting results that illustrate the applicability and superiority of the results obtained. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
11 pages, 1703 KiB  
Article
Synchronization of the Glycolysis Reaction-Diffusion Model via Linear Control Law
by Adel Ouannas, Iqbal M. Batiha, Stelios Bekiros, Jinping Liu, Hadi Jahanshahi, Ayman A. Aly and Abdulaziz H. Alghtani
Entropy 2021, 23(11), 1516; https://doi.org/10.3390/e23111516 - 15 Nov 2021
Cited by 12 | Viewed by 1894
Abstract
The Selkov system, which is typically employed to model glycolysis phenomena, unveils some rich dynamics and some other complex formations in biochemical reactions. In the present work, the synchronization problem of the glycolysis reaction-diffusion model is handled and examined. In addition, a novel [...] Read more.
The Selkov system, which is typically employed to model glycolysis phenomena, unveils some rich dynamics and some other complex formations in biochemical reactions. In the present work, the synchronization problem of the glycolysis reaction-diffusion model is handled and examined. In addition, a novel convenient control law is designed in a linear form and, on the other hand, the stability of the associated error system is demonstrated through utilizing a suitable Lyapunov function. To illustrate the applicability of the proposed schemes, several numerical simulations are performed in one- and two-spatial dimensions. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
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21 pages, 1238 KiB  
Article
Trapezoidal-Type Inequalities for Strongly Convex and Quasi-Convex Functions via Post-Quantum Calculus
by Humaira Kalsoom, Miguel Vivas-Cortez and Muhammad Amer Latif
Entropy 2021, 23(10), 1238; https://doi.org/10.3390/e23101238 - 22 Sep 2021
Cited by 4 | Viewed by 2003
Abstract
In this paper, we establish new (p,q)κ1-integral and (p,q)κ2-integral identities. By employing these new identities, we establish new (p,q)κ1 and [...] Read more.
In this paper, we establish new (p,q)κ1-integral and (p,q)κ2-integral identities. By employing these new identities, we establish new (p,q)κ1 and (p,q)κ2- trapezoidal integral-type inequalities through strongly convex and quasi-convex functions. Finally, some examples are given to illustrate the investigated results. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
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13 pages, 931 KiB  
Article
An Analytical Technique, Based on Natural Transform to Solve Fractional-Order Parabolic Equations
by Ravi P. Agarwal, Fatemah Mofarreh, Rasool Shah, Waewta Luangboon and Kamsing Nonlaopon
Entropy 2021, 23(8), 1086; https://doi.org/10.3390/e23081086 - 21 Aug 2021
Cited by 37 | Viewed by 2548
Abstract
This research article is dedicated to solving fractional-order parabolic equations using an innovative analytical technique. The Adomian decomposition method is well supported by natural transform to establish closed form solutions for targeted problems. The procedure is simple, attractive and is preferred over other [...] Read more.
This research article is dedicated to solving fractional-order parabolic equations using an innovative analytical technique. The Adomian decomposition method is well supported by natural transform to establish closed form solutions for targeted problems. The procedure is simple, attractive and is preferred over other methods because it provides a closed form solution for the given problems. The solution graphs are plotted for both integer and fractional-order, which shows that the obtained results are in good contact with the exact solution of the problems. It is also observed that the solution of fractional-order problems are convergent to the solution of integer-order problem. In conclusion, the current technique is an accurate and straightforward approximate method that can be applied to solve other fractional-order partial differential equations. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
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15 pages, 298 KiB  
Article
Some Parameterized Quantum Midpoint and Quantum Trapezoid Type Inequalities for Convex Functions with Applications
by Suphawat Asawasamrit, Muhammad Aamir Ali, Sotiris K. Ntouyas and Jessada Tariboon
Entropy 2021, 23(8), 996; https://doi.org/10.3390/e23080996 - 31 Jul 2021
Cited by 3 | Viewed by 1597
Abstract
Quantum information theory, an interdisciplinary field that includes computer science, information theory, philosophy, cryptography, and entropy, has various applications for quantum calculus. Inequalities and entropy functions have a strong association with convex functions. In this study, we prove quantum midpoint type inequalities, quantum [...] Read more.
Quantum information theory, an interdisciplinary field that includes computer science, information theory, philosophy, cryptography, and entropy, has various applications for quantum calculus. Inequalities and entropy functions have a strong association with convex functions. In this study, we prove quantum midpoint type inequalities, quantum trapezoidal type inequalities, and the quantum Simpson’s type inequality for differentiable convex functions using a new parameterized q-integral equality. The newly formed inequalities are also proven to be generalizations of previously existing inequities. Finally, using the newly established inequalities, we present some applications for quadrature formulas. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
21 pages, 543 KiB  
Article
Some New Hermite–Hadamard and Related Inequalities for Convex Functions via (p,q)-Integral
by Miguel Vivas-Cortez, Muhammad Aamir Ali, Hüseyin Budak, Humaira Kalsoom and Praveen Agarwal
Entropy 2021, 23(7), 828; https://doi.org/10.3390/e23070828 - 29 Jun 2021
Cited by 38 | Viewed by 2497
Abstract
In this investigation, for convex functions, some new (p,q)–Hermite–Hadamard-type inequalities using the notions of (p,q)π2 derivative and (p,q)π2 integral are obtained. Furthermore, for [...] Read more.
In this investigation, for convex functions, some new (p,q)–Hermite–Hadamard-type inequalities using the notions of (p,q)π2 derivative and (p,q)π2 integral are obtained. Furthermore, for (p,q)π2-differentiable convex functions, some new (p,q) estimates for midpoint and trapezoidal-type inequalities using the notions of (p,q)π2 integral are offered. It is also shown that the newly proved results for p=1 and q1 can be converted into some existing results. Finally, we discuss how the special means can be used to address newly discovered inequalities. Full article
(This article belongs to the Special Issue Advanced Numerical Methods for Differential Equations)
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