Special Issue "Numerical and Analytical Methods for Differential Equations and Systems"

A special issue of Fractal and Fractional (ISSN 2504-3110). This special issue belongs to the section "Numerical and Computational Methods".

Deadline for manuscript submissions: closed (30 June 2022) | Viewed by 11902

Special Issue Editors

Dr. Burcu Gürbüz
E-Mail Website
Guest Editor
Institute of Mathematics, Johannes-Gutenberg-University Mainz, Staudingerweg 9, 55128 Mainz, Germany
Interests: dynamical systems; differential equations; qualitative analysis; mathematical biology; numerical methods; programming
Dr. Arran Fernandez
E-Mail Website
Guest Editor
Department of Mathematics, Faculty of Arts and Sciences, Eastern Mediterranean University, Gazimagusa, TRNC, via Mersin 10, Turkey
Interests: fractional calculus; fractional differential equations; Mittag-Leffler functions; zeta functions; asymptotic analysis
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The theory and applications of differential equations have played an essential role both in the development of mathematics and in exploring new horizons in science. From a theoretical viewpoint, the qualitative theory of differential equations as well as analytical methods have contributed to the development of many new mathematical ideas and methodologies for solving ordinary and partial differential equations as well as systems of differential equations. From the viewpoint of applications, differential equations are crucially important for modeling any kind of dynamical systems or processes in real life. Even when analytical methods cannot be used, numerical methods are crucially important for investigating differential equations and understanding their structure to obtain approximations for their solutions.

Differential equations enable mathematics to be associated with other disciplines such as science, medicine, and engineering, since real-life problems in these fields give rise to differential equations which can only be solved using mathematics. Topics related to the theoretical and numerical aspects of differential equations have been undergoing tremendous development for decades. Numerical investigations in particular have played a decisive role in dynamical systems, control theory, and optimization, to name but a few areas. Important applications of differential equations have been found in physics, biology, chemistry, medicine, and engineering, and more recently in the development of new technologies. Therefore, the investigation and application of numerical and analytical methods to modern problems will be of immense value in many fields of study.

The aim of this Special Issue is to gather a collection of high-quality articles reflecting the current state of the art in the abovementioned topics. We welcome review and research papers covering any interesting developments related to these topics.

Dr. Burcu Gürbüz
Dr. Arran Fernandez
Guest Editors

Manuscript Submission Information

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Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 1800 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.

Keywords

  • Fractional differential equations
  • Dynamical systems
  • Delay differential equations
  • Ordinary differential equations
  • Partial differential equations
  • Analytical methods for differential equations
  • Numerical methods for differential equations
  • Stability and convergence analysis
  • Orthogonal polynomials
  • Functional differential equations
  • Qualitative theory of differential equations

Published Papers (17 papers)

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Research

Article
Multi-Solitons, Multi-Breathers and Multi-Rational Solutions of Integrable Extensions of the Kadomtsev–Petviashvili Equation in Three Dimensions
Fractal Fract. 2022, 6(8), 425; https://doi.org/10.3390/fractalfract6080425 - 31 Jul 2022
Viewed by 260
Abstract
The celebrated Korteweg–de Vries and Kadomtsev–Petviashvili (KP) equations are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. The question of constructing integrable evolution equations in three-spatial dimensions has been one of the most important open problems in the [...] Read more.
The celebrated Korteweg–de Vries and Kadomtsev–Petviashvili (KP) equations are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. The question of constructing integrable evolution equations in three-spatial dimensions has been one of the most important open problems in the history of integrability. Here, we study an integrable extension of the KP equation in three-spatial dimensions, which can be derived using a specific reduction of the integrable generalization of the KP equation in four-spatial and two-temporal dimensions derived in (Phys. Rev. Lett. 96, (2006) 190201). For this new integrable extension of the KP equation, we construct smooth multi-solitons, high-order breathers, and high-order rational solutions, by using Hirota’s bilinear method. Full article
Article
Dynamic Behavior Investigation of a Novel Epidemic Model Based on COVID-19 Risk Area Categorization
Fractal Fract. 2022, 6(8), 410; https://doi.org/10.3390/fractalfract6080410 - 26 Jul 2022
Viewed by 198
Abstract
This study establishes a compartment model for the categorized COVID-19 risk area. In this model, the compartments represent administrative regions at different transmission risk levels instead of individuals in traditional epidemic models. The county-level regions are partitioned into High-risk (H), Medium-risk (M), and [...] Read more.
This study establishes a compartment model for the categorized COVID-19 risk area. In this model, the compartments represent administrative regions at different transmission risk levels instead of individuals in traditional epidemic models. The county-level regions are partitioned into High-risk (H), Medium-risk (M), and Low-risk (L) areas dynamically according to the current number of confirmed cases. These risk areas are communicable by the movement of individuals. An LMH model is established with ordinary differential equations (ODEs). The basic reproduction number R0 is derived for the transmission of risk areas to determine whether the pandemic is controlled. The stability of this LHM model is investigated by a Lyapunov function and Poincare–Bendixson theorem. We prove that the disease-free equilibrium (R0 < 1) is globally asymptotically stable and the disease will die out. The endemic equilibrium (R0 > 1) is locally and globally asymptotically stable, and the disease will become endemic. The numerical simulation and data analysis support the previous theoretical proofs. For the first time, the compartment model is applied to investigate the dynamics of the transmission of the COVID-19 risk area. This work should be of great value in the development of precision region-specific containment strategies. Full article
Article
Approximate Solution of Fractional Differential Equation by Quadratic Splines
Fractal Fract. 2022, 6(7), 369; https://doi.org/10.3390/fractalfract6070369 - 30 Jun 2022
Viewed by 286
Abstract
In this article, we consider approximate solutions by quadratic splines for a fractional differential equation with two Caputo fractional derivatives, the orders of which satisfy 1<α<2 and 0<β<1. Numerical computing schemes of the two [...] Read more.
In this article, we consider approximate solutions by quadratic splines for a fractional differential equation with two Caputo fractional derivatives, the orders of which satisfy 1<α<2 and 0<β<1. Numerical computing schemes of the two fractional derivatives based on quadratic spline interpolation function are derived. Then, the recursion scheme for numerical solutions and the quadratic spline approximate solution are generated. Two numerical examples are used to check the proposed method. Additionally, comparisons with the L1L2 numerical solutions are conducted. For the considered fractional differential equation with the leading order α, the involved undetermined parameters in the quadratic spline interpolation function can be exactly resolved. Full article
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Article
Variable Step Hybrid Block Method for the Approximation of Kepler Problem
Fractal Fract. 2022, 6(6), 343; https://doi.org/10.3390/fractalfract6060343 - 20 Jun 2022
Cited by 1 | Viewed by 417
Abstract
In this article, a variable step size strategy is adopted in formulating a new variable step hybrid block method (VSHBM) for the solution of the Kepler problem, which is known to be a rigid and stiff differential equation. To derive the VSHBM, the [...] Read more.
In this article, a variable step size strategy is adopted in formulating a new variable step hybrid block method (VSHBM) for the solution of the Kepler problem, which is known to be a rigid and stiff differential equation. To derive the VSHBM, the step size ratio r is left the same, halved, or doubled in order to optimize the total number of steps, minimize the number of formulae stored in the code, and ensure that the method is zero-stable. The method is formulated by integrating the Lagrange polynomial with limits of integration selected at special points. The article further analyzed the stability, order, consistency, and convergence properties of the VSHBM. The stability regions of the VSHBM at different values of the step size ratios were also plotted and plots showed that the method is fit for solving the Kepler problem. The results generated were then compared with some existing methods, including the MATLAB inbuilt stiff solver (ode 15 s), with respect to total number of failure steps, total number of steps, total function calls, maximum error, and computation time. Full article
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Article
Analytical and Numerical Solutions for a Kind of High-Dimensional Fractional Order Equation
Fractal Fract. 2022, 6(6), 338; https://doi.org/10.3390/fractalfract6060338 - 17 Jun 2022
Viewed by 403
Abstract
In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. It is rarely studied because the order of the highest-order derivative term of this equation is higher than the common generalized (4+1)-dimensional Fokas equation. Firstly, the (4+1)-dimensional time-fractional Fokas equation with the [...] Read more.
In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. It is rarely studied because the order of the highest-order derivative term of this equation is higher than the common generalized (4+1)-dimensional Fokas equation. Firstly, the (4+1)-dimensional time-fractional Fokas equation with the Riemann–Liouville fractional derivative is derived by the semi-inverse method and variational method. Further, the symmetry of the time-fractional equation is obtained by the fractional Lie symmetry analysis method. Based on the symmetry, the conservation laws of the time fractional equation are constructed by the new conservation theorem. Then, the GG-expansion method is used here to solve the equation and obtain the exact traveling wave solutions. Finally, the spectral method in the spatial direction and the Gru¨nwald–Letnikov method in the time direction are considered to obtain the numerical solutions of the time-fractional equation. The numerical solutions are compared with the exact solutions, and the error results confirm the effectiveness of the proposed numerical method. Full article
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Article
Even-Order Neutral Delay Differential Equations with Noncanonical Operator: New Oscillation Criteria
Fractal Fract. 2022, 6(6), 313; https://doi.org/10.3390/fractalfract6060313 - 02 Jun 2022
Viewed by 432
Abstract
The main objective of our paper is to investigate the oscillatory properties of solutions of differential equations of neutral type and in the noncanonical case. We follow an approach that simplifies and extends the related previous results. Our results are an extension and [...] Read more.
The main objective of our paper is to investigate the oscillatory properties of solutions of differential equations of neutral type and in the noncanonical case. We follow an approach that simplifies and extends the related previous results. Our results are an extension and reflection of developments in the study of second-order equations. We also derive criteria for improving conditions that exclude the decreasing positive solutions of the considered equation. Full article
Article
Mixed Convection Flow over an Elastic, Porous Surface with Viscous Dissipation: A Robust Spectral Computational Approach
Fractal Fract. 2022, 6(5), 263; https://doi.org/10.3390/fractalfract6050263 - 10 May 2022
Cited by 1 | Viewed by 647
Abstract
A novel computational approach is developed to investigate the mixed convection, boundary layer flow over a nonlinear elastic (stretching or shrinking) surface. The viscous fluid is electrically conducting, incompressible, and propagating through a porous medium. The consequences of viscous dissipation, Joule heating, and [...] Read more.
A novel computational approach is developed to investigate the mixed convection, boundary layer flow over a nonlinear elastic (stretching or shrinking) surface. The viscous fluid is electrically conducting, incompressible, and propagating through a porous medium. The consequences of viscous dissipation, Joule heating, and heat sink/source of the volumetric rate of heat generation are also included in the energy balance equation. In order to formulate the mathematical modeling, a similarity analysis is performed. The numerical solution of nonlinear differential equations is accomplished through the use of a robust computational approach, which is identified as the Spectral Local Linearization Method (SLLM). The computational findings reported in this study show that, in addition to being simple to establish and numerically implement, the proposed method is very reliable in that it converges rapidly to achieve a specified goal and is more effective in resolving very complex models of nonlinear boundary value problems. In order to ensure the convergence of the proposed SLLM method, the Gauss–Seidel approach is used. The SLLM’s reliability and numerical stability can be optimized even more using Gauss–Seidel approach. The computational results for different emerging parameters are computed to show the behavior of velocity profile, skin friction coefficient, temperature profile, and Nusselt number. To evaluate the accuracy and the convergence of the obtained results, a comparison between the proposed approach and the bvp4c (built-in command in Matlab) method is presented. The Matlab software, which is used to generate machine time for executing the SLLM code, is also displayed in a table. Full article
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Article
Weighted Fractional Calculus: A General Class of Operators
Fractal Fract. 2022, 6(4), 208; https://doi.org/10.3390/fractalfract6040208 - 07 Apr 2022
Cited by 3 | Viewed by 624
Abstract
We conduct a formal study of a particular class of fractional operators, namely weighted fractional calculus, and its extension to the more general class known as weighted fractional calculus with respect to functions. We emphasise the importance of the conjugation relationships with the [...] Read more.
We conduct a formal study of a particular class of fractional operators, namely weighted fractional calculus, and its extension to the more general class known as weighted fractional calculus with respect to functions. We emphasise the importance of the conjugation relationships with the classical Riemann–Liouville fractional calculus, and use them to prove many fundamental properties of these operators. As examples, we consider special cases such as tempered, Hadamard-type, and Erdélyi–Kober operators. We also define appropriate modifications of the Laplace transform and convolution operations, and solve some ordinary differential equations in the setting of these general classes of operators. Full article
Article
Why Controlling the Asymptomatic Infection Is Important: A Modelling Study with Stability and Sensitivity Analysis
Fractal Fract. 2022, 6(4), 197; https://doi.org/10.3390/fractalfract6040197 - 31 Mar 2022
Cited by 1 | Viewed by 702
Abstract
The large proportion of asymptomatic patients is the major cause leading to the COVID-19 pandemic which is still a significant threat to the whole world. A six-dimensional ODE system (SEIAQR epidemical model) is established to study the dynamics of COVID-19 spreading considering infection [...] Read more.
The large proportion of asymptomatic patients is the major cause leading to the COVID-19 pandemic which is still a significant threat to the whole world. A six-dimensional ODE system (SEIAQR epidemical model) is established to study the dynamics of COVID-19 spreading considering infection by exposed, infected, and asymptomatic cases. The basic reproduction number derived from the model is more comprehensive including the contribution from the exposed, infected, and asymptomatic patients. For this more complex six-dimensional ODE system, we investigate the global and local stability of disease-free equilibrium, as well as the endemic equilibrium, whereas most studies overlooked asymptomatic infection or some other virus transmission features. In the sensitivity analysis, the parameters related to the asymptomatic play a significant role not only in the basic reproduction number R0. It is also found that the asymptomatic infection greatly affected the endemic equilibrium. Either in completely eradicating the disease or achieving a more realistic goal to reduce the COVID-19 cases in an endemic equilibrium, the importance of controlling the asymptomatic infection should be emphasized. The three-dimensional phase diagrams demonstrate the convergence point of the COVID-19 spreading under different initial conditions. In particular, massive infections will occur as shown in the phase diagram quantitatively in the case R0>1. Moreover, two four-dimensional contour maps of Rt are given varying with different parameters, which can offer better intuitive instructions on the control of the pandemic by adjusting policy-related parameters. Full article
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Article
A Reliable Approach for Solving Delay Fractional Differential Equations
Fractal Fract. 2022, 6(2), 124; https://doi.org/10.3390/fractalfract6020124 - 21 Feb 2022
Cited by 2 | Viewed by 610
Abstract
In this paper, we study a class of second-order delay fractional differential equations with a variable-order Caputo derivative. This type of equation is an extension to ordinary delay equations which are used in the modeling of several biological systems such as population dynamics, [...] Read more.
In this paper, we study a class of second-order delay fractional differential equations with a variable-order Caputo derivative. This type of equation is an extension to ordinary delay equations which are used in the modeling of several biological systems such as population dynamics, epidemiology, and immunology. Usually, fractional differential equations are difficult to solve analytically, and with fractional derivatives of variable-order, they become more challenging. Therefore, the need for reliable numerical techniques is worth investigating. To solve this type of equation, we derive a new approach based on the operational matrix. We use the shifted Chebyshev polynomials of the second kind as the basis for the approximate solutions. A convergence analysis is discussed and the uniform convergence of the approximate solutions is proven. Several examples are discussed to illustrate the efficiency of the presented approach. The computed errors, figures, and tables show that the approximate solutions converge to the exact ones by considering only a few terms in the expansion, and illustrate the novelty of the presented approach. Full article
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Article
Fixed Point Results for F-Contractions in Cone Metric Spaces over Topological Modules and Applications to Integral Equations
Fractal Fract. 2022, 6(1), 16; https://doi.org/10.3390/fractalfract6010016 - 29 Dec 2021
Viewed by 300
Abstract
In this paper, the concept of F-contraction was generalized for cone metric spaces over topological left modules and some fixed point results were obtained for self-mappings satisfying a contractive condition of this type. Some applications of the main result to the study [...] Read more.
In this paper, the concept of F-contraction was generalized for cone metric spaces over topological left modules and some fixed point results were obtained for self-mappings satisfying a contractive condition of this type. Some applications of the main result to the study of the existence and uniqueness of the solutions for certain types of integral equations were presented in the last part of the article, one of them being a fractional integral equation. Full article
Article
Numerical Simulation of Fractional Delay Differential Equations Using the Operational Matrix of Fractional Integration for Fractional-Order Taylor Basis
Fractal Fract. 2022, 6(1), 10; https://doi.org/10.3390/fractalfract6010010 - 26 Dec 2021
Viewed by 1207
Abstract
In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The [...] Read more.
In this paper, we consider numerical solutions for a general form of fractional delay differential equations (FDDEs) with fractional derivatives defined in the Caputo sense. A fractional integration operational matrix, created using a fractional Taylor basis, is applied to solve these FDDEs. The main characteristic of this approach is, by utilizing the operational matrix of fractional integration, to reduce the given differential equation to a set of algebraic equations with unknown coefficients. This equation system can be solved efficiently using a computer algorithm. A bound on the error for the best approximation and fractional integration are also given. Several examples are given to illustrate the validity and applicability of the technique. The efficiency of the presented method is revealed by comparing results with some existing solutions, the findings of some other approaches from the literature and by plotting absolute error figures. Full article
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Article
Asymptotic Separation of Solutions to Fractional Stochastic Multi-Term Differential Equations
Fractal Fract. 2021, 5(4), 256; https://doi.org/10.3390/fractalfract5040256 - 04 Dec 2021
Cited by 2 | Viewed by 682
Abstract
In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial [...] Read more.
In this paper, we study the exact asymptotic separation rate of two distinct solutions of Caputo stochastic multi-term differential equations (Caputo SMTDEs). Our goal in this paper is to establish results of the global existence and uniqueness and continuity dependence of the initial values of the solutions to Caputo SMTDEs with non-permutable matrices of order α(12,1) and β(0,1) whose coefficients satisfy a standard Lipschitz condition. For this class of systems, we then show the asymptotic separation property between two different solutions of Caputo SMTDEs with a more general condition based on λ. Furthermore, the asymptotic separation rate for the two distinct mild solutions reveals that our asymptotic results are general. Full article
Article
Approximating Real-Life BVPs via Chebyshev Polynomials’ First Derivative Pseudo-Galerkin Method
Fractal Fract. 2021, 5(4), 165; https://doi.org/10.3390/fractalfract5040165 - 12 Oct 2021
Cited by 4 | Viewed by 561
Abstract
An efficient technique, called pseudo-Galerkin, is performed to approximate some types of linear/nonlinear BVPs. The core of the performance process is the two well-known weighted residual methods, collocation and Galerkin. A novel basis of functions, consisting of first derivatives of Chebyshev polynomials, has [...] Read more.
An efficient technique, called pseudo-Galerkin, is performed to approximate some types of linear/nonlinear BVPs. The core of the performance process is the two well-known weighted residual methods, collocation and Galerkin. A novel basis of functions, consisting of first derivatives of Chebyshev polynomials, has been used. Consequently, new operational matrices for derivatives of any integer order have been introduced. An error analysis is performed to ensure the convergence of the presented method. In addition, the accuracy and the efficiency are verified by solving BVPs examples, including real-life problems. Full article
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Article
Oscillation Criteria of Solutions of Fourth-Order Neutral Differential Equations
Fractal Fract. 2021, 5(4), 155; https://doi.org/10.3390/fractalfract5040155 - 07 Oct 2021
Cited by 1 | Viewed by 517
Abstract
In this paper, we study the oscillation of solutions of fourth-order neutral delay differential equations in non-canonical form. By using Riccati transformation, we establish some new oscillation conditions. We provide some examples to examine the applicability of our results. Full article
Article
Advancement of Non-Newtonian Fluid with Hybrid Nanoparticles in a Convective Channel and Prabhakar’s Fractional Derivative—Analytical Solution
Fractal Fract. 2021, 5(3), 99; https://doi.org/10.3390/fractalfract5030099 - 17 Aug 2021
Cited by 6 | Viewed by 1038
Abstract
The present paper deals with the advancement of non-Newtonian fluid containing some nanoparticles between two parallel plates. A novel fractional operator is used to model memory effects, and analytical solutions are obtained for temperature and velocity fields by the method of Laplace transform. [...] Read more.
The present paper deals with the advancement of non-Newtonian fluid containing some nanoparticles between two parallel plates. A novel fractional operator is used to model memory effects, and analytical solutions are obtained for temperature and velocity fields by the method of Laplace transform. Moreover, a parametric study is elaborated to show the impact of flow parameters and presented in graphical form. As a result, dual solutions are predicted for increasing values of fractional parameters for short and long times. Furthermore, by increasing nanoparticle concentration, the temperature can be raised along with decreasing velocity. A fractional approach can provide new insight for the analytical solutions which makes the interpretation of the results easier and enable the way of testing possible approximate solutions. Full article
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Article
Qualitative Behavior of Unbounded Solutions of Neutral Differential Equations of Third-Order
Fractal Fract. 2021, 5(3), 95; https://doi.org/10.3390/fractalfract5030095 - 12 Aug 2021
Cited by 1 | Viewed by 944
Abstract
New oscillatory properties for the oscillation of unbounded solutions to a class of third-order neutral differential equations with several deviating arguments are established. Several oscillation results are established by using generalized Riccati transformation and a integral average technique under the case of unbounded [...] Read more.
New oscillatory properties for the oscillation of unbounded solutions to a class of third-order neutral differential equations with several deviating arguments are established. Several oscillation results are established by using generalized Riccati transformation and a integral average technique under the case of unbounded neutral coefficients. Examples are given to prove the significance of new theorems. Full article
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