Special Issue "Applications of Partial Differential Equations in Engineering"

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Difference and Differential Equations".

Deadline for manuscript submissions: closed (31 December 2021).

Special Issue Editor

Prof. Dr. Francisco Ureña
E-Mail Website
Guest Editor
Escuela Técnica Superior de Ingenieros Industriales, National Distance Education University, Madrid, Spain
Interests: partial differential equations; applied mathematics; computational mathematics; numerical analysis; discrete math; mechanics

Special Issue Information

Dear Colleagues,

Partial differential equations have become one extensive topic in Mathematics, Physics and Engineering due to the novel techniques recently developed and the great achievements in Computational Sciences. Both theoretical and applied viewpoints have obtained great attention from many different natural sciences.

A partial list of topics includes modeling; solution techniques and applications of computational methods in a variety of areas (e.g., liquid and gas dynamics, solid and structural mechanics, bio-mechanics, etc.); variational formulations and numerical algorithms related to implementation of the finite and boundary element methods; finite difference and finite volume methods; and other basic computational methodologies

Prof. Dr. Francisco Ureña
Guest Editor

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Keywords

  • Applied mathematics
  • Modeling
  • Fluid mechanics
  • Computational methods
  • Finite elements
  • Finite differences
  • Solution techniques
  • Mesh generation
  • Computational engineering

Published Papers (21 papers)

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Research

Article
A Note on a Meshless Method for Fractional Laplacian at Arbitrary Irregular Meshes
Mathematics 2021, 9(22), 2843; https://doi.org/10.3390/math9222843 - 10 Nov 2021
Viewed by 332
Abstract
The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using [...] Read more.
The existence and uniqueness of the discrete solutions of a porous medium equation with diffusion are demonstrated. The Cauchy problem contains a fractional Laplacian and it is equivalent to the extension formulation in the sense of trace and harmonic extension operators. By using the generalized finite difference method, we obtain the convergence of the numerical solution to the classical/theoretical solution of the equation for nonnegative initial data sufficiently smooth and bounded. This procedure allows us to use meshes with complicated geometry (more realistic) or with an irregular distribution of nodes (providing more accurate solutions where needed). Some numerical results are presented in arbitrary irregular meshes to illustrate the potential of the method. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Mixed Mesh Finite Volume Method for 1D Hyperbolic Systems with Application to Plug-Flow Heat Exchangers
Mathematics 2021, 9(20), 2609; https://doi.org/10.3390/math9202609 - 16 Oct 2021
Viewed by 327
Abstract
We present a finite volume method formulated on a mixed Eulerian-Lagrangian mesh for highly advective 1D hyperbolic systems altogether with its application to plug-flow heat exchanger modeling/simulation. Advection of sharp moving fronts is an important problem in fluid dynamics, and even a simple [...] Read more.
We present a finite volume method formulated on a mixed Eulerian-Lagrangian mesh for highly advective 1D hyperbolic systems altogether with its application to plug-flow heat exchanger modeling/simulation. Advection of sharp moving fronts is an important problem in fluid dynamics, and even a simple transport equation cannot be solved precisely by having a finite number of nodes/elements/volumes. Finite volume methods are known to introduce numerical diffusion, and there exist a wide variety of schemes to minimize its occurrence; the most recent being adaptive grid methods such as moving mesh methods or adaptive mesh refinement methods. We present a solution method for a class of hyperbolic systems with one nonzero time-dependent characteristic velocity. This property allows us to rigorously define a finite volume method on a grid that is continuously moving by the characteristic velocity (Lagrangian grid) along a static Eulerian grid. The advective flux of the flowing field is, by this approach, removed from cell-to-cell interactions, and the ability to advect sharp fronts is therefore enhanced. The price to pay is a fixed velocity-dependent time sampling and a time delay in the solution. For these reasons, the method is best suited for systems with a dominating advection component. We illustrate the method’s properties on an illustrative advection-decay equation example and a 1D plug flow heat exchanger. Such heat exchanger model can then serve as a convection-accurate dynamic model in estimation and control algorithms for which it was developed. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Adaptive Boundary Control for a Certain Class of Reaction–Advection–Diffusion System
Mathematics 2021, 9(18), 2224; https://doi.org/10.3390/math9182224 - 10 Sep 2021
Viewed by 670
Abstract
Several phenomena in nature are subjected to the interaction of various physical parameters, which, if these latter are well known, allow us to predict the behavior of such phenomena. In most cases, these physical parameters are not exactly known, or even more these [...] Read more.
Several phenomena in nature are subjected to the interaction of various physical parameters, which, if these latter are well known, allow us to predict the behavior of such phenomena. In most cases, these physical parameters are not exactly known, or even more these are unknown, so identification techniques could be employed in order to estimate their values. Systems for which their inputs and outputs vary both temporally and spatially are the so-called distributed parameter systems (DPSs) modeled through partial differential equations (PDEs). The way in which their parameters evolve with respect to time may not always be known and may also induce undesired behavior of the dynamics of the system. To reverse the above, the well-known adaptive boundary control technique can be used to estimate the unknown parameters assuring a stable behavior of the dynamics of the system. In this work, we focus our attention on the design of an adaptive boundary control for a parabolic type reaction–advection–diffusion PDE under the assumption of unknown parameters for both advection and reaction terms and Robin and Neumann boundary conditions. An identifier PDE system is established and parameter update laws are designed following the certainty equivalence approach with a passive identifier. The performance of the adaptive Neumann stabilizing controller is validated via numerical simulation. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Convective Heat Transfer of a Hybrid Nanofluid over a Nonlinearly Stretching Surface with Radiation Effect
Mathematics 2021, 9(18), 2220; https://doi.org/10.3390/math9182220 - 10 Sep 2021
Cited by 2 | Viewed by 365
Abstract
The flow of the hybrid nanofluid (copper–titanium dioxide/water) over a nonlinearly stretching surface was studied with suction and radiation effect. The governing partial differential equations were then converted into non-linear ordinary differential equations by using proper similarity transformations. Therefore, these equations were solved [...] Read more.
The flow of the hybrid nanofluid (copper–titanium dioxide/water) over a nonlinearly stretching surface was studied with suction and radiation effect. The governing partial differential equations were then converted into non-linear ordinary differential equations by using proper similarity transformations. Therefore, these equations were solved by applying a numerical technique, namely Chebyshev pseudo spectral differentiation matrix. The results of the flow field, temperature distribution, reduced skin friction coefficient and reduced Nusselt number were deduced. It was found that the rising of the mass flux parameter slows down the velocity and, hence, decreases the temperature. Further, on enlarging the stretching parameter, the velocity and temperature increases and decreases, respectively. In addition, it was mentioned that the radiation parameter can effectively control the thermal boundary layer. Finally, the temperature decreases when the values of the temperature parameter increases. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Reliability Sampling Design for the Lifetime Performance Index of Gompertz Lifetime Distribution under Progressive Type I Interval Censoring
Mathematics 2021, 9(17), 2109; https://doi.org/10.3390/math9172109 - 31 Aug 2021
Cited by 1 | Viewed by 405
Abstract
In this artificial intelligence era, the constantly changing technology makes production techniques become sophisticated and complicated. Therefore, manufacturers are dedicated to improving the quality of products by increasing the lifetime in order to achieve the quality standards demanded by consumers. For products with [...] Read more.
In this artificial intelligence era, the constantly changing technology makes production techniques become sophisticated and complicated. Therefore, manufacturers are dedicated to improving the quality of products by increasing the lifetime in order to achieve the quality standards demanded by consumers. For products with lifetime following a Gompertz distribution, the lifetime performance index was used to measure the performance of manufacturing process under progressive type I interval censoring. The sampling design is investigated to reach the given level of significance and power level. When inspection interval length is fixed and the number of inspection intervals is not fixed, the required number of inspection intervals and sample size with minimum total cost are determined and tabulated. When the termination time is not fixed, the required number of inspection intervals, sample size, and equal interval length reaching the minimum total cost are determined and tabulated. The optimal parameter values are tabulated for the practical use of users. Finally, one practical example is given for the illustrative aim to show the implementation of this sampling design to collect data and the collected data are used to conduct the testing procedure to see if the process is capable. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
On a Novel Numerical Scheme for Riesz Fractional Partial Differential Equations
Mathematics 2021, 9(16), 2014; https://doi.org/10.3390/math9162014 - 23 Aug 2021
Cited by 2 | Viewed by 355
Abstract
In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability [...] Read more.
In this paper, we consider numerical solutions for Riesz space fractional partial differential equations with a second order time derivative. We propose a Galerkin finite element scheme for both the temporal and spatial discretizations. For the proposed numerical scheme, we derive sharp stability estimates as well as optimal a priori error estimates. Extensive numerical experiments are conducted to verify the promising features of the newly proposed method. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
Article
On a Class of Second-Order PDE&PDI Constrained Robust Modified Optimization Problems
Mathematics 2021, 9(13), 1473; https://doi.org/10.3390/math9131473 - 23 Jun 2021
Cited by 3 | Viewed by 470
Abstract
In this paper, by using scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi-time controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional scalar variational control problems with mixed [...] Read more.
In this paper, by using scalar multiple integral cost functionals and the notion of convexity associated with a multiple integral functional driven by an uncertain multi-time controlled second-order Lagrangian, we develop a new mathematical framework on multi-dimensional scalar variational control problems with mixed constraints implying second-order partial differential equations (PDEs) and inequations (PDIs). Concretely, we introduce and investigate an auxiliary (modified) variational control problem, which is much easier to study, and provide some equivalence results by using the notion of a normal weak robust optimal solution. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
Article
Convergence and Numerical Solution of a Model for Tumor Growth
Mathematics 2021, 9(12), 1355; https://doi.org/10.3390/math9121355 - 11 Jun 2021
Cited by 1 | Viewed by 580
Abstract
In this paper, we show the application of the meshless numerical method called “Generalized Finite Diference Method” (GFDM) for solving a model for tumor growth with nutrient density, extracellular matrix and matrix degrading enzymes, [recently proposed by Li and Hu]. We derive the [...] Read more.
In this paper, we show the application of the meshless numerical method called “Generalized Finite Diference Method” (GFDM) for solving a model for tumor growth with nutrient density, extracellular matrix and matrix degrading enzymes, [recently proposed by Li and Hu]. We derive the discretization of the parabolic–hyperbolic–parabolic–elliptic system by means of the explicit formulae of the GFDM. We provide a theoretical proof of the convergence of the spatial–temporal scheme to the continuous solution and we show several examples over regular and irregular distribution of points. This shows the feasibility of the method for solving this nonlinear model appearing in Biology and Medicine in complicated and realistic domains. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Four-Quadrant Riemann Problem for a 2×2 System II
Mathematics 2021, 9(6), 592; https://doi.org/10.3390/math9060592 - 10 Mar 2021
Viewed by 648
Abstract
In previous work, we considered a four-quadrant Riemann problem for a 2×2 hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, [...] Read more.
In previous work, we considered a four-quadrant Riemann problem for a 2×2 hyperbolic system in which delta shock appears at the initial discontinuity without assuming that each jump of the initial data projects exactly one plane elementary wave. In this paper, we consider the case that does not involve a delta shock at the initial discontinuity. We classified 18 topologically distinct solutions and constructed analytic and numerical solutions for each case. The constructed analytic solutions show the rich structure of wave interactions in the Riemann problem, which coincide with the computed numerical solutions. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Conventional Partial and Complete Solutions of the Fundamental Equations of Fluid Mechanics in the Problem of Periodic Internal Waves with Accompanying Ligaments Generation
Mathematics 2021, 9(6), 586; https://doi.org/10.3390/math9060586 - 10 Mar 2021
Cited by 2 | Viewed by 486
Abstract
The problem of generating beams of periodic internal waves in a viscous, exponentially stratified fluid by a band oscillating along an inclined plane is considered by the methods of the theory of singular perturbations in the linear and weakly nonlinear approximations. The complete [...] Read more.
The problem of generating beams of periodic internal waves in a viscous, exponentially stratified fluid by a band oscillating along an inclined plane is considered by the methods of the theory of singular perturbations in the linear and weakly nonlinear approximations. The complete solution to the linear problem, which satisfies the boundary conditions on the emitting surface, is constructed taking into account the previously proposed classification of flow structural components described by complete solutions of the linearized system of fundamental equations without involving additional force or mass sources. Analyses includes all components satisfying the dispersion relation that are periodic waves and thin accompanying ligaments, the transverse scale of which is determined by the kinematic viscosity and the buoyancy frequency. Ligaments are located both near the emitting surface and in the bulk of the liquid in the form of wave beam envelopes. Calculations show that in a nonlinear description of all components, both waves and ligaments interact directly with each other in all combinations: waves-waves, waves-ligaments, and ligaments-ligaments. Direct interactions of the components that generate new harmonics of internal waves occur despite the differences in their scales. Additionally, the problem of generating internal waves by a rapidly bi-harmonically oscillating vertical band is considered. If the difference in the frequencies of the spectral components of the band movement is less than the buoyancy frequency, the nonlinear interacting ligaments generate periodic waves as well. The estimates made show that the amplitudes of such waves are large enough to be observed under laboratory conditions. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Four-Quadrant Riemann Problem for a 2 × 2 System Involving Delta Shock
Mathematics 2021, 9(2), 138; https://doi.org/10.3390/math9020138 - 10 Jan 2021
Cited by 1 | Viewed by 1071
Abstract
In this paper, a four-quadrant Riemann problem for a 2×2 system of hyperbolic conservation laws is considered in the case of delta shock appearing at the initial discontinuity. We also remove the restriction in that there is only one planar wave [...] Read more.
In this paper, a four-quadrant Riemann problem for a 2×2 system of hyperbolic conservation laws is considered in the case of delta shock appearing at the initial discontinuity. We also remove the restriction in that there is only one planar wave at each initial discontinuity. We include the existence of two elementary waves at each initial discontinuity and classify 14 topologically distinct solutions. For each case, we construct an analytic solution and compute the numerical solution. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Multiple Comparisons for Exponential Median Lifetimes with the Control Based on Doubly Censored Samples
Mathematics 2021, 9(1), 76; https://doi.org/10.3390/math9010076 - 31 Dec 2020
Viewed by 462
Abstract
Under doubly censoring, the one-stage multiple comparison procedures with the control in terms of exponential median lifetimes are presented. The uniformly minimum variance unbiased estimator for median lifetime is found. The upper bounds, lower bounds and two-sided confidence intervals for the difference between [...] Read more.
Under doubly censoring, the one-stage multiple comparison procedures with the control in terms of exponential median lifetimes are presented. The uniformly minimum variance unbiased estimator for median lifetime is found. The upper bounds, lower bounds and two-sided confidence intervals for the difference between each median lifetimes and the median lifetime of the control population are developed. Statistical tables of critical values are constructed for the practical use of our proposed procedures. Users can use these simultaneous confidence intervals to determine whether the performance of treatment populations is better than or worse than the control population in agriculture and pharmaceutical industries. At last, one practical example is provided to illustrate the proposed procedures. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
Article
Towards a Vector Field Based Approach to the Proper Generalized Decomposition (PGD)
Mathematics 2021, 9(1), 34; https://doi.org/10.3390/math9010034 - 25 Dec 2020
Viewed by 562
Abstract
A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best [...] Read more.
A novel algorithm called the Proper Generalized Decomposition (PGD) is widely used by the engineering community to compute the solution of high dimensional problems. However, it is well-known that the bottleneck of its practical implementation focuses on the computation of the so-called best rank-one approximation. Motivated by this fact, we are going to discuss some of the geometrical aspects of the best rank-one approximation procedure. More precisely, our main result is to construct explicitly a vector field over a low-dimensional vector space and to prove that we can identify its stationary points with the critical points of the best rank-one optimization problem. To obtain this result, we endow the set of tensors with fixed rank-one with an explicit geometric structure. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
Article
Complex Ginzburg–Landau Equation with Generalized Finite Differences
Mathematics 2020, 8(12), 2248; https://doi.org/10.3390/math8122248 - 20 Dec 2020
Viewed by 694
Abstract
In this paper we obtain a novel implementation for irregular clouds of nodes of the meshless method called Generalized Finite Difference Method for solving the complex Ginzburg–Landau equation. We derive the explicit formulae for the spatial derivative and an explicit scheme by splitting [...] Read more.
In this paper we obtain a novel implementation for irregular clouds of nodes of the meshless method called Generalized Finite Difference Method for solving the complex Ginzburg–Landau equation. We derive the explicit formulae for the spatial derivative and an explicit scheme by splitting the equation into a system of two parabolic PDEs. We prove the conditional convergence of the numerical scheme towards the continuous solution under certain assumptions. We obtain a second order approximation as it is clear from the numerical results. Finally, we provide several examples of its application over irregular domains in order to test the accuracy of the explicit scheme, as well as comparison with other numerical methods. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
State Feedback Regulation Problem to the Reaction-Diffusion Equation
Mathematics 2020, 8(11), 1983; https://doi.org/10.3390/math8111983 - 06 Nov 2020
Viewed by 952
Abstract
In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded [...] Read more.
In this work, we explore the state feedback regulator problem (SFRP) in order to achieve the goal for trajectory tracking with harmonic disturbance rejection to one-dimensional (1-D) reaction-diffusion (R-D) equation, namely, a partial differential equation of parabolic type, while taking into account bounded input, output, and disturbance operators, a finite-dimensional exosystem (exogenous system), and the state of the exosystem as the state to the feedback law. As is well-known, the SFRP can be solved only if the so-called Francis (regulator) equations have solution. In our work, we try with the solution of the Francis equations from the 1-D R-D equation following given criteria to the eigenvalues from the exosystem and transfer function of the system, but the state operator is here defined in terms of the Sturm–Liouville differential operator (SLDO). Within this framework, the SFRP is then solved for the 1-D R-D equation. The numerical simulation results validate the performance of the regulator. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
A Meshless Method Based on the Laplace Transform for the 2D Multi-Term Time Fractional Partial Integro-Differential Equation
Mathematics 2020, 8(11), 1972; https://doi.org/10.3390/math8111972 - 06 Nov 2020
Cited by 2 | Viewed by 613
Abstract
In this article, we propose a localized transform based meshless method for approximating the solution of the 2D multi-term partial integro-differential equation involving the time fractional derivative in Caputo’s sense with a weakly singular kernel. The purpose of coupling the localized meshless method [...] Read more.
In this article, we propose a localized transform based meshless method for approximating the solution of the 2D multi-term partial integro-differential equation involving the time fractional derivative in Caputo’s sense with a weakly singular kernel. The purpose of coupling the localized meshless method with the Laplace transform is to avoid the time stepping procedure by eliminating the time variable. Then, we utilize the local meshless method for spatial discretization. The solution of the original problem is obtained as a contour integral in the complex plane. In the literature, numerous contours are available; in our work, we will use the recently introduced improved Talbot contour. We approximate the contour integral using the midpoint rule. The bounds of stability for the differentiation matrix of the scheme are derived, and the convergence is discussed. The accuracy, efficiency, and stability of the scheme are validated by numerical experiments. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Boundary Control for a Certain Class of Reaction-Advection-Diffusion System
Mathematics 2020, 8(11), 1854; https://doi.org/10.3390/math8111854 - 22 Oct 2020
Cited by 1 | Viewed by 1025
Abstract
There are physical phenomena, involving diffusion and structural vibrations, modeled by partial differential equations (PDEs) whose solution reflects their spatial distribution. Systems whose dynamics evolve on an infinite-dimensional Hilbert space, i.e., infinite-dimensional systems, are modeled by PDEs. The aim when designing a controller [...] Read more.
There are physical phenomena, involving diffusion and structural vibrations, modeled by partial differential equations (PDEs) whose solution reflects their spatial distribution. Systems whose dynamics evolve on an infinite-dimensional Hilbert space, i.e., infinite-dimensional systems, are modeled by PDEs. The aim when designing a controller for infinite-dimensional systems is similar to that for finite-dimensional systems, i.e., the control system must be stable. Another common goal is to design the controller in such a way that the response of the system does not be affected by external disturbances. The controller design for finite-dimensional systems is not an easy task, so, the controller design for infinite-dimensional systems is even more challenging. The backstepping control approach is a dominant methodology for boundary feedback design. In this work, we try with the backstepping design for the boundary control of a reaction-advection-diffusion (R-A-D) equation, namely, a type parabolic PDE, but with constant coefficients and Neumann boundary conditions, with actuation in one of these latter. The heat equation with Neumann boundary conditions is considered as the target system. Dynamics of the open- and closed-loop solution of the PDE system are validated via numerical simulation. The MATLAB®-based numerical algorithm related with the implementation of the control scheme is here included. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
The Application of Accurate Exponential Solution of a Differential Equation in Optimizing Stability Control of One Class of Chaotic System
by and
Mathematics 2020, 8(10), 1740; https://doi.org/10.3390/math8101740 - 10 Oct 2020
Cited by 1 | Viewed by 627
Abstract
For many nonlinear systems in our life, the chaos phenomenon generated under certain conditions in special cases will split the system and result in a crash-down of the system. This paper discusses the stable control of one class of chaotic systems and a [...] Read more.
For many nonlinear systems in our life, the chaos phenomenon generated under certain conditions in special cases will split the system and result in a crash-down of the system. This paper discusses the stable control of one class of chaotic systems and a control method based on the accurate exponential solution of a differential equation is used. Compared with other methods, the advantages are: this method determines that the system can exponentially converge at the origin and the convergence rate can be easily regulated. The chaotic system with unknown parameters is also deduced and validated by using this method. In practical application, it is found that the ship’s electric system also has the same model, so it has certain practical significance. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
Closed-Form Solutions and Conserved Vectors of a Generalized (3+1)-Dimensional Breaking Soliton Equation of Engineering and Nonlinear Science
Mathematics 2020, 8(10), 1692; https://doi.org/10.3390/math8101692 - 01 Oct 2020
Cited by 4 | Viewed by 637
Abstract
In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie [...] Read more.
In this article, we examine a (3+1)-dimensional generalized breaking soliton equation which is highly applicable in the fields of engineering and nonlinear sciences. Closed-form solutions in the form of Jacobi elliptic functions of the underlying equation are derived by the method of Lie symmetry reductions together with direct integration. Moreover, the (G/G)-expansion technique is engaged, which consequently guarantees closed-form solutions of the equation structured in the form of trigonometric and hyperbolic functions. In addition, we secure a power series analytical solution of the underlying equation. Finally, we construct local conserved vectors of the aforementioned equation by employing two approaches: the general multiplier method and Ibragimov’s theorem. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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Article
One-Stage Multiple Comparisons with the Control for Exponential Median Lifetimes under Heteroscedasticity
Mathematics 2020, 8(9), 1405; https://doi.org/10.3390/math8091405 - 21 Aug 2020
Cited by 1 | Viewed by 491
Abstract
When the additional sample for the second stage may not be available, one-stage multiple comparisons for exponential median lifetimes with the control under heteroscedasticity including one-sided and two-sided confidence intervals are proposed in this paper since the median is a more robust measure [...] Read more.
When the additional sample for the second stage may not be available, one-stage multiple comparisons for exponential median lifetimes with the control under heteroscedasticity including one-sided and two-sided confidence intervals are proposed in this paper since the median is a more robust measure of central tendency compared to the mean. These intervals can be used to identify treatment populations that are better than the control or worse than the control in terms of median lifetimes in agriculture, stock market, pharmaceutical industries. Tables of critical values are obtained for practical use. An example of comparing the survival days for four categories of lung cancer in a standard chemotherapeutic agent is given to demonstrate the proposed procedures. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
Article
On Generalized Fourier’s and Fick’s Laws in Bio-Convection Flow of Magnetized Burgers’ Nanofluid Utilizing Motile Microorganisms
Mathematics 2020, 8(7), 1186; https://doi.org/10.3390/math8071186 - 19 Jul 2020
Cited by 2 | Viewed by 829
Abstract
This article describes the features of bio-convection and motile microorganisms in magnetized Burgers’ nanoliquid flows by stretchable sheet. Theory of Cattaneo–Christov mass and heat diffusions is also discussed. The Buongiorno phenomenon for nanoliquid motion in a Burgers’ fluid is employed in view of [...] Read more.
This article describes the features of bio-convection and motile microorganisms in magnetized Burgers’ nanoliquid flows by stretchable sheet. Theory of Cattaneo–Christov mass and heat diffusions is also discussed. The Buongiorno phenomenon for nanoliquid motion in a Burgers’ fluid is employed in view of the Cattaneo–Christov relation. The control structure of governing partial differential equations (PDEs) is changed into appropriate ordinary differential equations (ODEs) by suitable transformations. To get numerical results of nonlinear systems, the bvp4c solver provided in the commercial software MATLAB is employed. Numerical and graphical data for velocity, temperature, nanoparticles concentration and microorganism profiles are obtained by considering various estimations of prominent physical parameters. Our computations depict that the temperature field has direct relation with the thermal Biot number and Burgers’ fluid parameter. Here, temperature field is enhanced for growing estimations of thermal Biot number and Burgers’ fluid parameter. Full article
(This article belongs to the Special Issue Applications of Partial Differential Equations in Engineering)
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