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Mathematics
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Computational Methods in Applied Analysis and Mathematical Modeling
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Computational Methods in Applied Analysis and Mathematical Modeling

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (31 December 2019) | Viewed by 20454

Special Issue Editors

Prof. Dr. Juan Luis García Guirao

E-Mail Website
Guest Editor
Department of Applied Mathematics and Statistics, Technical University of Cartagena, 30203 Cartagena, Spain
Interests: dynamical systems; celestial mechanics; Hamiltonian systems; low-dimensional dynamics; differential equations
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Mustafa Bayram

E-Mail Website
Guest Editor
Graduate School of Natural and Applied Sciences Istanbul Gelisim University Avcilar-Istanbul, Turkey
Interests: application of the computer algebra techniques to the enzymes kinetics; numerical solution of differential-algebraic equation; metabolic control theory; mathematical modeling; numerical solution of the ordinary and partial differential equations

Special Issue Information

Dear Colleagues,

Nowadays, computational methods play a very important role in engineering mathematics. Based upon extensive applications in sciences such as physics, mechanics, chemistry, and biology, research on ordinary or partial differential equations, their systems, and other relative topics are active and widespread in the engineering world. This Special Issue will cover topics of high current interest falling within the scope of mathematics and welcomes research papers of the highest quality. The objective of this Special Issue is to emphasise the eminency of computational methods and their applications. Potential topics include modeling, solution techniques, and applications of computational methods in a variety of areas, variational formulations, and numerical algorithms related to implementation of the finite and boundary element methods, collocation methods, finite difference and finite volume methods, and other basic computational methodologies, but are not limited to the following:

  • Mathematical and computer modelling;
  • Computational methods for real world problems;
  • Applied mechanics in engineering;
  • Mathematical analysis of computational and theoretical models;
  • The application of mathematical models in biological and chemical sciences;
  • New methods for solving differential equations arising in engineering problems;
  • Fractional calculus in engineering;
  • Controllability of systems of differential equations or numerical methods applied to the solutions of differential equations arising in engineering mathematics;
  • Iteration methods for solving partial and ordinary differential equations;
  • Numerical functional analysis and applications;
  • Local and nonlocal boundary value problems for partial differential equations;
  • Stochastic partial differential equations and applications;
  • Computational methods in partial differential equations;
  • Fuzzy linear equations and their solutions;
  • Linear programming.

Prof. Dr. Mustafa Bayram
Prof. Dr. Juan Luis García Guirao
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Published Papers (8 papers)

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Research

9 pages, 1039 KiB  
Article
Regarding New Wave Patterns of the Newly Extended Nonlinear (2+1)-Dimensional Boussinesq Equation with Fourth Order
by Juan Luis García Guirao, Haci Mehmet Baskonus and Ajay Kumar
Mathematics 2020, 8(3), 341; https://doi.org/10.3390/math8030341 - 04 Mar 2020
Cited by 34 | Viewed by 2849
Abstract
This paper applies the sine-Gordon expansion method to the extended nonlinear (2+1)-dimensional Boussinesq equation. Many new dark, complex and mixed dark-bright soliton solutions of the governing model are derived. Moreover, for better understanding of the results, 2D, 3D and contour graphs under the [...] Read more.
This paper applies the sine-Gordon expansion method to the extended nonlinear (2+1)-dimensional Boussinesq equation. Many new dark, complex and mixed dark-bright soliton solutions of the governing model are derived. Moreover, for better understanding of the results, 2D, 3D and contour graphs under the strain conditions and the suitable values of parameters are also plotted. Full article
(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)
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26 pages, 421 KiB  
Article
Modified Accelerated Bundle-Level Methods and Their Application in Two-Stage Stochastic Programming
by Chunming Tang, Bo He and Zhenzhen Wang
Mathematics 2020, 8(2), 265; https://doi.org/10.3390/math8020265 - 17 Feb 2020
Cited by 1 | Viewed by 1574
Abstract
The accelerated prox-level (APL) and uniform smoothing level (USL) methods recently proposed by Lan (Math Program, 149: 1–45, 2015) can achieve uniformly optimal complexity when solving black-box convex programming (CP) and structure non-smooth CP problems. In this paper, we propose two modified accelerated [...] Read more.
The accelerated prox-level (APL) and uniform smoothing level (USL) methods recently proposed by Lan (Math Program, 149: 1–45, 2015) can achieve uniformly optimal complexity when solving black-box convex programming (CP) and structure non-smooth CP problems. In this paper, we propose two modified accelerated bundle-level type methods, namely, the modified APL (MAPL) and modified USL (MUSL) methods. Compared with the original APL and USL methods, the MAPL and MUSL methods reduce the number of subproblems by one in each iteration, thereby improving the efficiency of the algorithms. Conclusions of optimal iteration complexity of the proposed algorithms are established. Furthermore, the modified methods are applied to the two-stage stochastic programming, and numerical experiments are implemented to illustrate the advantages of our methods in terms of efficiency and accuracy. Full article
(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)
17 pages, 800 KiB  
Article
On C-To-R-Based Iteration Methods for a Class of Complex Symmetric Weakly Nonlinear Equations
by Min-Li Zeng and Guo-Feng Zhang
Mathematics 2020, 8(2), 208; https://doi.org/10.3390/math8020208 - 06 Feb 2020
Viewed by 1559
Abstract
To avoid solving the complex systems, we first rewrite the complex-valued nonlinear system to real-valued form (C-to-R) equivalently. Then, based on separable property of the linear and the nonlinear terms, we present a C-to-R-based Picard iteration method and a nonlinear C-to-R-based splitting (NC-to-R) [...] Read more.
To avoid solving the complex systems, we first rewrite the complex-valued nonlinear system to real-valued form (C-to-R) equivalently. Then, based on separable property of the linear and the nonlinear terms, we present a C-to-R-based Picard iteration method and a nonlinear C-to-R-based splitting (NC-to-R) iteration method for solving a class of large sparse and complex symmetric weakly nonlinear equations. At each inner process iterative step of the new methods, one only needs to solve the real subsystems with the same symmetric positive and definite coefficient matrix. Therefore, the computational workloads and computational storage will be saved in actual implements. The conditions for guaranteeing the local convergence are studied in detail. The quasi-optimal parameters are also proposed for both the C-to-R-based Picard iteration method and the NC-to-R iteration method. Numerical experiments are performed to show the efficiency of the new methods. Full article
(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)
20 pages, 646 KiB  
Article
Approximate Solutions for Fractional Boundary Value Problems via Green-CAS Wavelet Method
by Muhammad Ismail, Umer Saeed, Jehad Alzabut and Mujeeb ur Rehman
Mathematics 2019, 7(12), 1164; https://doi.org/10.3390/math7121164 - 02 Dec 2019
Cited by 24 | Viewed by 2544
Abstract
In this study, we present a novel numerical scheme for the approximate solutions of linear as well as non-linear ordinary differential equations of fractional order with boundary conditions. This method combines Cosine and Sine (CAS) wavelets together with Green function, called Green-CAS method. [...] Read more.
In this study, we present a novel numerical scheme for the approximate solutions of linear as well as non-linear ordinary differential equations of fractional order with boundary conditions. This method combines Cosine and Sine (CAS) wavelets together with Green function, called Green-CAS method. The method simplifies the existing CAS wavelet method and does not require conventional operational matrices of integration for certain cases. Quasilinearization technique is used to transform non-linear fractional differential equations to linear equations and then Green-CAS method is applied. Furthermore, the proposed method has also been analyzed for convergence, particularly in the context of error analysis. Sufficient conditions for the existence of unique solutions are established for the boundary value problem under consideration. Moreover, to elaborate the effectiveness and accuracy of the proposed method, results of essential numerical applications have also been documented in graphical as well as tabular form. Full article
(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)
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14 pages, 276 KiB  
Article
Approximate Conservation Laws of Nonvariational Differential Equations
by Sameerah Jamal
Mathematics 2019, 7(7), 574; https://doi.org/10.3390/math7070574 - 27 Jun 2019
Cited by 7 | Viewed by 2079
Abstract
The concept of an approximate multiplier (integrating factor) is introduced. Such multipliers are shown to give rise to approximate local conservation laws for differential equations that admit a small perturbation. We develop an explicit, algorithmic and efficient method to construct both the approximate [...] Read more.
The concept of an approximate multiplier (integrating factor) is introduced. Such multipliers are shown to give rise to approximate local conservation laws for differential equations that admit a small perturbation. We develop an explicit, algorithmic and efficient method to construct both the approximate multipliers and their corresponding approximate fluxes. Our method is applicable to equations with any number of independent and dependent variables, linear or nonlinear, is adaptable to deal with any order of perturbation and does not require the existence of a variational principle. Several important perturbed equations are presented to exemplify the method, such as the approximate KdV equation. Finally, a second treatment of approximate multipliers is discussed. Full article
(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)
15 pages, 1095 KiB  
Article
The Gegenbauer Wavelets-Based Computational Methods for the Coupled System of Burgers’ Equations with Time-Fractional Derivative
by Neslihan Ozdemir, Aydin Secer and Mustafa Bayram
Mathematics 2019, 7(6), 486; https://doi.org/10.3390/math7060486 - 28 May 2019
Cited by 24 | Viewed by 2845
Abstract
In this study, Gegenbauer wavelets are used to present two numerical methods for solving the coupled system of Burgers’ equations with a time-fractional derivative. In the presented methods, we combined the operational matrix of fractional integration with the Galerkin method and the collocation [...] Read more.
In this study, Gegenbauer wavelets are used to present two numerical methods for solving the coupled system of Burgers’ equations with a time-fractional derivative. In the presented methods, we combined the operational matrix of fractional integration with the Galerkin method and the collocation method to obtain a numerical solution of the coupled system of Burgers’ equations with a time-fractional derivative. The properties of Gegenbauer wavelets were used to transform this system to a system of nonlinear algebraic equations in the unknown expansion coefficients. The Galerkin method and collocation method were used to find these coefficients. The main aim of this study was to indicate that the Gegenbauer wavelets-based methods is suitable and efficient for the coupled system of Burgers’ equations with time-fractional derivative. The obtained results show the applicability and efficiency of the presented Gegenbaur wavelets-based methods. Full article
(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)
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6 pages, 227 KiB  
Article
Some Remarks on a Variational Method for Stiff Differential Equations
by Sergio Amat, María José Legaz and Pablo Pedregal
Mathematics 2019, 7(5), 455; https://doi.org/10.3390/math7050455 - 20 May 2019
Cited by 1 | Viewed by 2015
Abstract
We have recently proposed a variational framework for the approximation of systems of differential equations. We associated, in a natural way, with the original problem, a certain error functional. The discretization is based on standard descent schemes, and we can use a variable-step [...] Read more.
We have recently proposed a variational framework for the approximation of systems of differential equations. We associated, in a natural way, with the original problem, a certain error functional. The discretization is based on standard descent schemes, and we can use a variable-step implementation. The minimization problem has a unique solution, and the approach has a global convergence. The use of our error-functional strategy was considered by other authors, but using a completely different way to derive the discretization. Their technique was based on the use of an integral form of the Euler equation for a related optimal control problem, combined with an adapted version of the shooting method, and the cyclic coordinate descent method. In this note, we illustrate and compare our strategy to theirs from a numerical point of view. Full article
(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)
18 pages, 2797 KiB  
Article
Bifurcation and Stability Analysis of a System of Fractional-Order Differential Equations for a Plant–Herbivore Model with Allee Effect
by Ali Yousef and Fatma Bozkurt Yousef
Mathematics 2019, 7(5), 454; https://doi.org/10.3390/math7050454 - 20 May 2019
Cited by 19 | Viewed by 3650
Abstract
This article concerns establishing a system of fractional-order differential equations (FDEs) to model a plant–herbivore interaction. Firstly, we show that the model has non-negative solutions, and then we study the existence and stability analysis of the constructed model. To investigate the case according [...] Read more.
This article concerns establishing a system of fractional-order differential equations (FDEs) to model a plant–herbivore interaction. Firstly, we show that the model has non-negative solutions, and then we study the existence and stability analysis of the constructed model. To investigate the case according to a low population density of the plant population, we incorporate the Allee function into the model. Considering the center manifold theorem and bifurcation theory, we show that the model shows flip bifurcation. Finally, the simulation results agree with the theoretical studies. Full article
(This article belongs to the Special Issue Computational Methods in Applied Analysis and Mathematical Modeling)
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