Research

14 pages, 349 KiB

Open AccessArticle

Detecting Line Sources inside Cylinders by Analytical Algorithms

by Dimitrios S. Lazaridis and Nikolaos L. Tsitsas

Mathematics 2023, 11(13), 2935; https://doi.org/10.3390/math11132935 - 30 Jun 2023

Cited by 1 | Viewed by 623

Inverse problems for line sources radiating inside a homogeneous magneto-dielectric cylinder are investigated. The developed algorithms concern the determination of the location and the current of each source. These algorithms are mostly analytical and are based on proper exploitation of the moments obtained [...] Read more.

Inverse problems for line sources radiating inside a homogeneous magneto-dielectric cylinder are investigated. The developed algorithms concern the determination of the location and the current of each source. These algorithms are mostly analytical and are based on proper exploitation of the moments obtained by integrating the product of the total field on the cylindrical boundary with complex exponential functions. The information on the unknown parameters of the problem is encoded in these moments, and hence all parameters can be recovered by means of relatively simple explicit expressions. The cases of one and two sources are considered and analyzed. Under certain conditions, the permittivity and permeability of the cylinder are also recovered. The results from two types of numerical experiments are presented: (i) for a single source, the effect of noise on the boundary data is studied, (ii) for two sources, the pertinent nonlinear system of equations is solved numerically and the accuracy of the derived solution is discussed. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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15 pages, 1259 KiB

Open AccessArticle

Mathematical Modeling of Brain Swelling in Electroencephalography and Magnetoencephalography

by Athena Papargiri, George Fragoyiannis and Vasileios S. Kalantonis

Mathematics 2023, 11(11), 2582; https://doi.org/10.3390/math11112582 - 05 Jun 2023

Viewed by 1168

Abstract

In the present paper, the forward problem of EEG and MEG is discussed, where the head is modeled by a spherical two-shell piecewise-homogeneous conductor with a neuronal current source positioned in the exterior shell area representing the brain tissue, while the interior shell [...] Read more.

In the present paper, the forward problem of EEG and MEG is discussed, where the head is modeled by a spherical two-shell piecewise-homogeneous conductor with a neuronal current source positioned in the exterior shell area representing the brain tissue, while the interior shell portrays a cerebral edema. We consider constant conductivity, which assumes different values in each compartment, where the expansions of the electric potential and the magnetic field are represented via spherical harmonics. Furthermore, we demonstrate the reduction of our analytical results to the single-compartment model while it is shown that the magnetic field in the exterior of the conductor is a function only of the dipole moment and its position. Consequently, it does not depend on the inhomogeneity dictated by the interior shell, a fact that verifies the efficiency of the model. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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20 pages, 3240 KiB

Open AccessArticle

Optical Solitons and Modulation Instability Analysis with Lakshmanan–Porsezian–Daniel Model Having Parabolic Law of Self-Phase Modulation

by Kaltham K. Al-Kalbani, Khalil S. Al-Ghafri, Edamana V. Krishnan and Anjan Biswas

Mathematics 2023, 11(11), 2471; https://doi.org/10.3390/math11112471 - 27 May 2023

Cited by 2 | Viewed by 767

Abstract

This paper seeks to find optical soliton solutions for Lakshmanan–Porsezian–Daniel (LPD) model with the parabolic law of nonlinearity. The spatiotemporal dispersion is included to the model, as it can contribute to handling the problem of internet bottleneck. This study was performed analytically using [...] Read more.

This paper seeks to find optical soliton solutions for Lakshmanan–Porsezian–Daniel (LPD) model with the parabolic law of nonlinearity. The spatiotemporal dispersion is included to the model, as it can contribute to handling the problem of internet bottleneck. This study was performed analytically using the traveling wave hypothesis to reduce the model to an integrable form. Then, the resulting equation was handled with two approaches, namely, the auxiliary equation method and the Bernoulli subordinary differential equation (sub-ODE) method. With an intentional focus on hyperbolic function solutions, abundant optical soliton waves including W-shaped, bright, dark, kink-dark, singular, kink, and antikink solitons were derived with the existing conditions. Furthermore, the behaviors of some optical solitons are illustrated. The spatiotemporal dispersion was found to significantly affect the pulse propagation dynamics. Finally, the modulation instability (MI) of the LPD model is explained in detail along with the extraction of the expression of MI gain. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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15 pages, 2122 KiB

Open AccessArticle

Non-Convex Particle-in-Cell Model for the Mathematical Study of the Microscopic Blood Flow

by Hadjinicolaou Maria and Eleftherios Protopapas

Mathematics 2023, 11(9), 2156; https://doi.org/10.3390/math11092156 - 04 May 2023

Viewed by 1165

Abstract

The field of fluid mechanics was further explored through the use of a particle-in-cell model for the mathematical study of the Stokes axisymmetric flow through a swarm of erythrocytes in a small vessel. The erythrocytes were modeled as inverted prolate spheroids encompassed by [...] Read more.

The field of fluid mechanics was further explored through the use of a particle-in-cell model for the mathematical study of the Stokes axisymmetric flow through a swarm of erythrocytes in a small vessel. The erythrocytes were modeled as inverted prolate spheroids encompassed by a fluid fictitious envelope. The fourth order partial differential equation governing the flow was completed with Happel-type boundary conditions which dictate no fluid slip on the inverted spheroid and a shear stress free non-permeable fictitious boundary. Through innovative means, such as the Kelvin inversion method and the R-semiseparation technique, a stream function was obtained as series expansion of Gegenbauer functions of the first and the second kinds of even order. Based on this, analytical expressions of meaningful hydrodynamic quantities, such as the velocity and the pressure field, were calculated and depicted in informative graphs. Using the first term of the stream function, the drag force exerted on the erythrocyte and the drag coefficient were calculated relative to the solid volume fraction of the cell. The results of the present research can be used for the further investigation of particle–fluid interactions. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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19 pages, 629 KiB

Open AccessArticle

The Gravity Force Generated by a Non-Rotating Level Ellipsoid of Revolution with Low Eccentricity as a Series of Spherical Harmonics

by Gerassimos Manoussakis

Mathematics 2023, 11(9), 1974; https://doi.org/10.3390/math11091974 - 22 Apr 2023

Cited by 1 | Viewed by 608

Abstract

The gravity force of a gravity field generated by a non-rotating level ellipsoid of revolution enclosing mass M is given as a solution of a partial differential equation along with a boundary condition of Dirichlet type. The partial differential equation is formulated herein [...] Read more.

The gravity force of a gravity field generated by a non-rotating level ellipsoid of revolution enclosing mass M is given as a solution of a partial differential equation along with a boundary condition of Dirichlet type. The partial differential equation is formulated herein on the basis of the behavior of spherical gravity fields. A classical solution to this equation is represented on the basis of spherical harmonics. The series representation of the solution is exploited in order to conduct a rigorous asymptotic analysis with respect to eccentricity. Finally, the Dirichlet boundary problem is solved for the case of an ellipsoid of revolution (spheroid) with low eccentricity. This has been accomplished on the basis of asymptotic analysis, which resulted in the determination of the coefficients participating in the spherical harmonics expansion. The limiting case of this series expresses the gravity force of a non-rotating sphere. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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16 pages, 577 KiB

Open AccessArticle

The Regulator Problem to the Convection–Diffusion Equation

by Andrés A. Ramírez and Francisco Jurado

Mathematics 2023, 11(8), 1944; https://doi.org/10.3390/math11081944 - 20 Apr 2023

Viewed by 1146

Abstract

In this paper, from linear operator, semigroup and Sturm–Liouville problem theories, an abstract system model for the convection–diffusion (C–D) equation is proposed. The state operator for this abstract system model is here defined as given in the form of the Sturm–Liouville differential operator [...] Read more.

In this paper, from linear operator, semigroup and Sturm–Liouville problem theories, an abstract system model for the convection–diffusion (C–D) equation is proposed. The state operator for this abstract system model is here defined as given in the form of the Sturm–Liouville differential operator (SLDO) plus an integral term of the same SLDO. Our aim is to achieve the trajectory tracking task in the presence of external disturbances to the C–D equation invoking the regulator problem theory, where the state from a finite-dimensional exosystem is the state to the feedback law. In this context, the regulator (Francis) equations, established from the abstract system model for the C–D equation, here are solved; i.e., the state feedback regulator problem (SFRP) for the C–D system has a solution. Our proposal is validated via numerical simulation results. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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11 pages, 312 KiB

Open AccessArticle

Nonlinear Stability of the Monotone Traveling Wave for the Isothermal Fluid Equations with Viscous and Capillary Terms

by Xiang Li, Weiguo Zhang and Haipeng Jin

Mathematics 2023, 11(7), 1734; https://doi.org/10.3390/math11071734 - 05 Apr 2023

Cited by 1 | Viewed by 891

Abstract

We prove the existence of the monotone traveling wave for the isothermal fluid equations with viscous and capillary terms by the planar dynamical system method. We obtain that the monotone traveling wave is asymptotically stable under the suitable perturbation. In the process of [...] Read more.

We prove the existence of the monotone traveling wave for the isothermal fluid equations with viscous and capillary terms by the planar dynamical system method. We obtain that the monotone traveling wave is asymptotically stable under the suitable perturbation. In the process of establishing the uniform a priori estimate, we dispose the capillary term reasonably according to the feature of the equations, and find the appropriate weighted function to overcome the difficulty caused by the non-convex pressure function. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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14 pages, 1009 KiB

Open AccessArticle

Solitary Wave Solution of a Generalized Fractional–Stochastic Nonlinear Wave Equation for a Liquid with Gas Bubbles

by Wael W. Mohammed, Farah M. Al-Askar, Clemente Cesarano and Mahmoud El-Morshedy

Mathematics 2023, 11(7), 1692; https://doi.org/10.3390/math11071692 - 01 Apr 2023

Cited by 4 | Viewed by 865

Abstract

In the sense of a conformable fractional operator, we consider a generalized fractional–stochastic nonlinear wave equation (GFSNWE). This equation may be used to depict several nonlinear physical phenomena occurring in a liquid containing gas bubbles. The analytical solutions of the GFSNWE are obtained [...] Read more.

In the sense of a conformable fractional operator, we consider a generalized fractional–stochastic nonlinear wave equation (GFSNWE). This equation may be used to depict several nonlinear physical phenomena occurring in a liquid containing gas bubbles. The analytical solutions of the GFSNWE are obtained by using the F-expansion and the Jacobi elliptic function methods with the Riccati equation. Due to the presence of noise and the conformable derivative, some solutions that were achieved are shown together with their physical interpretations. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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25 pages, 1669 KiB

Open AccessArticle

Study of Stochastic–Fractional Drinfel’d–Sokolov–Wilson Equation for M-Shaped Rational, Homoclinic Breather, Periodic and Kink-Cross Rational Solutions

by Shami A. M. Alsallami, Syed T. R. Rizvi and Aly R. Seadawy

Mathematics 2023, 11(6), 1504; https://doi.org/10.3390/math11061504 - 20 Mar 2023

Cited by 20 | Viewed by 1413

Abstract

We explore stochastic–fractional Drinfel’d–Sokolov–Wilson (SFDSW) equations for some wave solutions such as the cross-kink rational wave solution, periodic cross-rational wave solution and homoclinic breather wave solution. We also examine some M-shaped solutions such as the M-shaped rational solution, M-shaped rational solution with one [...] Read more.

We explore stochastic–fractional Drinfel’d–Sokolov–Wilson (SFDSW) equations for some wave solutions such as the cross-kink rational wave solution, periodic cross-rational wave solution and homoclinic breather wave solution. We also examine some M-shaped solutions such as the M-shaped rational solution, M-shaped rational solution with one and two kink waves. We also derive the M-shaped interaction with rogue and kink waves and the M-shaped interaction with periodic and kink waves. This model is used in mathematical physics, surface physics, plasma physics, population dynamics and applied sciences. Moreover, we also show our results graphically in different dimensions. We obtain these solutions under some constraint conditions. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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11 pages, 1011 KiB

Open AccessArticle

The Soliton Solutions of the Stochastic Shallow Water Wave Equations in the Sense of Beta-Derivative

by Wael W. Mohammed, Farah M. Al-Askar, Clemente Cesarano and Elkhateeb S. Aly

Mathematics 2023, 11(6), 1338; https://doi.org/10.3390/math11061338 - 09 Mar 2023

Cited by 11 | Viewed by 1228

Abstract

The stochastic shallow water wave equation (SSWWE) in the sense of the beta-derivative is considered in this study. The solutions of the SSWWE are obtained using the F-expansion technique with the Riccati equation and He’s semi-inverse method. Since the shallow water equation has [...] Read more.

The stochastic shallow water wave equation (SSWWE) in the sense of the beta-derivative is considered in this study. The solutions of the SSWWE are obtained using the F-expansion technique with the Riccati equation and He’s semi-inverse method. Since the shallow water equation has many uses in ocean engineering, including river irrigation flows, tidal waves, tsunami prediction, and weather simulations, the solutions discovered can be utilized to represent a wide variety of exciting physical events. We create many 2D and 3D graphs to demonstrate how the beta-derivative and Brownian motion affect the analytical solutions of the SSWWE. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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23 pages, 1562 KiB

Open AccessArticle

The Squeeze Film Effect with a High-Pressure Boundary in Aerostatic Bearings

by Yangong Wu, Jiadai Xue, Zheng Qiao, Wentao Chen and Bo Wang

Mathematics 2023, 11(3), 742; https://doi.org/10.3390/math11030742 - 01 Feb 2023

Viewed by 1450

Abstract

The squeeze film effect was discussed in several fields, but mostly under the same pressure boundary conditions. However, pressures at the inlet and outlet are different for aerostatic bearings. In this paper, the dynamic Reynolds equation group, with the stiffness and damping pressure [...] Read more.

The squeeze film effect was discussed in several fields, but mostly under the same pressure boundary conditions. However, pressures at the inlet and outlet are different for aerostatic bearings. In this paper, the dynamic Reynolds equation group, with the stiffness and damping pressure written separately, is deducted and numerically solved with a high-pressure boundary for a parallel flat and circular thin film. The circular thin film considers the two results of the supply pressure boundary inside and outside. All dynamic pressure distribution and stiffness curves are given in a dimensionless form, and a comparative analysis of squeeze film characteristics with and without external pressure is conducted. From the calculation results, it can be concluded that the squeeze effect shows damping for zero-frequency and stiffness for infinite-frequency for compressible lubricants. The dynamic pressure in the static high pressure region is also high at high frequencies affected by gas compressibility. Based on these analytical results, the transfer functions of the thin film are given to further analyze the dynamic performance of aerostatic bearings, and the shape of the response curve approximates an exponential decay form, even when the amplitude increases to 10% of the gas film thickness. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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25 pages, 415 KiB

Open AccessArticle

Reductions and Exact Solutions of Nonlinear Wave-Type PDEs with Proportional and More Complex Delays

by Andrei D. Polyanin and Vsevolod G. Sorokin

Mathematics 2023, 11(3), 516; https://doi.org/10.3390/math11030516 - 18 Jan 2023

Cited by 4 | Viewed by 1391

Abstract

The study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with [...] Read more.

The study gives a brief overview of publications on exact solutions for functional PDEs with delays of various types and on methods for constructing such solutions. For the first time, second-order wave-type PDEs with a nonlinear source term containing the unknown function with proportional time delay, proportional space delay, or both time and space delays are considered. In addition to nonlinear wave-type PDEs with constant speed, equations with variable speed are also studied. New one-dimensional reductions and exact solutions of such PDEs with proportional delay are obtained using solutions of simpler PDEs without delay and methods of separation of variables for nonlinear PDEs. Self-similar solutions, additive and multiplicative separable solutions, generalized separable solutions, and some other solutions are presented. More complex nonlinear functional PDEs with a variable time or space delay of general form are also investigated. Overall, more than thirty wave-type equations with delays that admit exact solutions are described. The study results can be used to test numerical methods and investigate the properties of the considered and related PDEs with proportional or more complex variable delays. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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20 pages, 443 KiB

Open AccessArticle

On a Novel Algorithmic Determination of Acoustic Low Frequency Coefficients for Arbitrary Impenetrable Scatterers

by Foteini Kariotou, Dimitris E. Sinikis and Maria Hadjinicolaou

Mathematics 2022, 10(23), 4487; https://doi.org/10.3390/math10234487 - 28 Nov 2022

Viewed by 908

Abstract

The calculation of low frequency expansions for acoustic wave scattering has been under thorough investigation for many decades due to their utility in technological applications. In the present work, we revisit the acoustic Low Frequency Scattering theory, and we provide the theoretical framework [...] Read more.

The calculation of low frequency expansions for acoustic wave scattering has been under thorough investigation for many decades due to their utility in technological applications. In the present work, we revisit the acoustic Low Frequency Scattering theory, and we provide the theoretical framework of a new algorithmic procedure for deriving the scattering coefficients of the total pressure field, produced by a plane wave excitation of an arbitrary, convex impenetrable scatterer. The proposed semi-analytical procedure reduces the demands for computation time and errors significantly since it includes mainly algebraic and linear integral operators. Based on the Atkinson–Wilcox theorem, any order low frequency scattering coefficient can be calculated, in finite steps, through algebraic operators at all steps, except for the last one, where a regular Fredholm integral equation with a continuous and separable integral kernel is needed to be solved. Explicit, ready to use formulae are provided for the first three low frequency scattering coefficients, demonstrating the applicability of the algorithm. The validation of the obtained formulae is demonstrated through recovering of the well-known analytical results for the case of a radially symmetric scatterer. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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39 pages, 672 KiB

Open AccessArticle

On the Global Behaviour of Solutions for a Delayed Viscoelastic-Type Petrovesky Wave Equation with p-Laplacian Operator and Logarithmic Source

by Bochra Belhadji, Jehad Alzabut, Mohammad Esmael Samei and Nahid Fatima

Mathematics 2022, 10(22), 4194; https://doi.org/10.3390/math10224194 - 09 Nov 2022

Cited by 4 | Viewed by 1247

Abstract

This research is concerned with a nonlinear p-Laplacian-type wave equation with a strong damping and logarithmic source term under the null Dirichlet boundary condition. We establish the global existence of the solutions by using the potential well method. Moreover, we prove the [...] Read more.

This research is concerned with a nonlinear p-Laplacian-type wave equation with a strong damping and logarithmic source term under the null Dirichlet boundary condition. We establish the global existence of the solutions by using the potential well method. Moreover, we prove the stability of the solutions by the Nakao technique. An example with illustrative figures is provided as an application. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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12 pages, 710 KiB

Open AccessArticle

Application of a Variant of Mountain Pass Theorem in Modeling Real Phenomena

by Irina Meghea

Mathematics 2022, 10(19), 3476; https://doi.org/10.3390/math10193476 - 23 Sep 2022

Cited by 2 | Viewed by 1180

Abstract

Mountain Pass Theorem (MPT) is an important result in variational methods with multiple applications in partial differential equations involved in mathematical physics. Starting from a variant of MPT, a new result concerning the existence of the solution for certain mathematical physics problems involving [...] Read more.

Mountain Pass Theorem (MPT) is an important result in variational methods with multiple applications in partial differential equations involved in mathematical physics. Starting from a variant of MPT, a new result concerning the existence of the solution for certain mathematical physics problems involving p-Laplacian and p-pseudo-Laplacian has been obtained. Based on the main theorem, the existence, possibly the uniqueness, and characterization of solutions for models such as nonlinear elastic membrane, glacier sliding, and pseudo torsion problem have been obtained. The novelty of the work consists of the formulation of the central result under weaker conditions requested by the chosen variant of MPT, the proof of this statement, and its application in solving above mentioned problems. While the expressions of such Dirichlet and/or von Neumann problems were already completed, this proposed solving method suggests some specific numerical methods to construct the appropriate solution. A general goal of this paper is the extension of the applicative pallet of this way to construct the solutions encountered in modeling real processes developed within new emerging technologies. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

17 pages, 712 KiB

Open AccessArticle

Advanced Analytic Self-Similar Solutions of Regular and Irregular Diffusion Equations

by Imre Ferenc Barna and László Mátyás

Mathematics 2022, 10(18), 3281; https://doi.org/10.3390/math10183281 - 09 Sep 2022

Cited by 5 | Viewed by 1304

Abstract

We study the diffusion equation with an appropriate change of variables. This equation is, in general, a partial differential equation (PDE). With the self-similar and related Ansatz, we transform the PDE of diffusion to an ordinary differential equation. The solutions of the PDE [...] Read more.

We study the diffusion equation with an appropriate change of variables. This equation is, in general, a partial differential equation (PDE). With the self-similar and related Ansatz, we transform the PDE of diffusion to an ordinary differential equation. The solutions of the PDE belong to a family of functions which are presented for the case of infinite horizon. In the presentation, we accentuate the physically reasonable solutions. We also study time-dependent diffusion phenomena, where the spreading may vary in time. To describe the process, we consider time-dependent diffusion coefficients. The obtained analytic solutions all can be expressed with Kummer’s functions. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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10 pages, 933 KiB

Open AccessArticle

Non-Relativistic Treatment of the 2D Electron System Interacting via Varshni–Shukla Potential Using the Asymptotic Iteration Method

by Collins Okon Edet, Salman Mahmoud, Etido P. Inyang, Norshamsuri Ali, Syed Alwee Aljunid, Rosdisham Endut, Akpan Ndem Ikot and Muhammad Asjad

Mathematics 2022, 10(15), 2824; https://doi.org/10.3390/math10152824 - 08 Aug 2022

Cited by 24 | Viewed by 1778

Abstract

The nonrelativistic treatment of the Varshni–Shukla potential (V–SP) in the presence of magnetic and Aharanov–Bohm fields is carried out using the asymptotic iteration method (AIM). The energy equation and wave function are derived analytically. The energy levels are summed to obtain the partition [...] Read more.

The nonrelativistic treatment of the Varshni–Shukla potential (V–SP) in the presence of magnetic and Aharanov–Bohm fields is carried out using the asymptotic iteration method (AIM). The energy equation and wave function are derived analytically. The energy levels are summed to obtain the partition function, which is employed to derive the expressions for the thermomagnetic properties of the V–SP. These properties are analyzed extensively using graphical representations. It is observed that in the various settings of the analysis, the system shows a diamagnetic characteristic, and the specific heat capacity behavior agrees with the recognized Dulong–Petit law, although some slight anomaly is observed. This irregular behavior could be attributed to a Schottky anomaly. Our findings will be valuable in a variety of fields of physics, including chemical, molecular and condensed matter physics, where our derived models could be applied to study other diatomic molecules and quantum dots, respectively. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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16 pages, 644 KiB

Open AccessArticle

A Fast Galerkin Approach for Solving the Fractional Rayleigh–Stokes Problem via Sixth-Kind Chebyshev Polynomials

by Ahmed Gamal Atta, Waleed Mohamed Abd-Elhameed, Galal Mahrous Moatimid and Youssri Hassan Youssri

Mathematics 2022, 10(11), 1843; https://doi.org/10.3390/math10111843 - 27 May 2022

Cited by 16 | Viewed by 1290

Abstract

Herein, a spectral Galerkin method for solving the fractional Rayleigh–Stokes problem involving a nonlinear source term is analyzed. Two kinds of basis functions that are related to the shifted sixth-kind Chebyshev polynomials are selected and utilized in the numerical treatment of the problem. [...] Read more.

Herein, a spectral Galerkin method for solving the fractional Rayleigh–Stokes problem involving a nonlinear source term is analyzed. Two kinds of basis functions that are related to the shifted sixth-kind Chebyshev polynomials are selected and utilized in the numerical treatment of the problem. Some specific integer and fractional derivative formulas are used to introduce our proposed numerical algorithm. Moreover, the stability and convergence accuracy are derived in detail. As a final validation of our theoretical results, we present a few numerical examples. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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12 pages, 276 KiB

Open AccessArticle

The Non-Linear Fokker–Planck Equation in Low-Regularity Space

by Yingzhe Fan and Bo Tang

Mathematics 2022, 10(9), 1576; https://doi.org/10.3390/math10091576 - 07 May 2022

Viewed by 1037

Abstract

We construct the global existence and exponential time decay rates of mild solutions to the non-linear Fokker–Planck equation near a global Maxwellians with small-amplitude initial data in the low regularity function space

L_{k}^{1} L_{T}^{\infty} L_{v}^{2}

where the [...] Read more.

We construct the global existence and exponential time decay rates of mild solutions to the non-linear Fokker–Planck equation near a global Maxwellians with small-amplitude initial data in the low regularity function space

L_{k}^{1} L_{T}^{\infty} L_{v}^{2}

where the regularity assumption on the initial data is weaker. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

15 pages, 2465 KiB

Open AccessArticle

Solitary Wave Solutions for the Higher Dimensional Jimo-Miwa Dynamical Equation via New Mathematical Techniques

by Aly R. Seadawy, Hanadi Zahed and Mujahid Iqbal

Mathematics 2022, 10(7), 1011; https://doi.org/10.3390/math10071011 - 22 Mar 2022

Cited by 13 | Viewed by 1365

Abstract

In this study, under the considerations of symbolic computation with the help of Mathematica software, various types of solitary wave solutions for the (3 + 1)-dimensional Jimo–Miwa (JM) equation are successfully constructed based on the extended modified rational expansion method. The constructed solutions [...] Read more.

In this study, under the considerations of symbolic computation with the help of Mathematica software, various types of solitary wave solutions for the (3 + 1)-dimensional Jimo–Miwa (JM) equation are successfully constructed based on the extended modified rational expansion method. The constructed solutions are novel and more general for the JM equation named kink wave solutions, anti-kink wave solutions, bright and dark solutions, mixed solutions in the shape of bright-dark solutions, and periodic waves, which do not exist in the existing literature. The physical phenomena of the demonstrated results is represented graphically by two-dimensional, three-dimensional, and contour images with the help of Mathematica. The obtained results will be widely used to explain the various interesting physical structures in the areas of optics, plasma, gas, acoustics, classical mechanics, fluid dynamics, heat transfer, and many others. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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14 pages, 289 KiB

Open AccessArticle

Continuous Dependence for the Boussinesq Equations under Reaction Boundary Conditions in R²

by Jincheng Shi and Yan Liu

Mathematics 2022, 10(6), 991; https://doi.org/10.3390/math10060991 - 19 Mar 2022

Viewed by 1071

Abstract

In this paper, we studied the continuous dependence result for the Boussinesq equations. We considered the case where

Ω

was a bounded domain in

R^{2}

. Temperatures T and C satisfied reaction boundary conditions. A first-order inequality for the differences of energy [...] Read more.

In this paper, we studied the continuous dependence result for the Boussinesq equations. We considered the case where

Ω

was a bounded domain in

R^{2}

. Temperatures T and C satisfied reaction boundary conditions. A first-order inequality for the differences of energy could be derived. An integration of this inequality produced a continuous dependence result. The result told us that the continuous dependence type stability was also valid for the Boussinesq coefficient

λ

of the Boussinesq equations with reaction boundary conditions. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

19 pages, 5620 KiB

Open AccessArticle

Novel Analytical Approach for the Space-Time Fractional (2+1)-Dimensional Breaking Soliton Equation via Mathematical Methods

by Abdulmohsen D. Alruwaili, Aly R. Seadawy, Asghar Ali and Sid Ahmed O. Beinane

Mathematics 2021, 9(24), 3253; https://doi.org/10.3390/math9243253 - 16 Dec 2021

Cited by 7 | Viewed by 1748

Abstract

The aim of this work is to build novel analytical wave solutions of the nonlinear space-time fractional (2+1)-dimensional breaking soliton equations, with regards to the modified Riemann–Liouville derivative, by employing mathematical schemes, namely, the improved simple equation and modified F-expansion methods. We used [...] Read more.

The aim of this work is to build novel analytical wave solutions of the nonlinear space-time fractional (2+1)-dimensional breaking soliton equations, with regards to the modified Riemann–Liouville derivative, by employing mathematical schemes, namely, the improved simple equation and modified F-expansion methods. We used the fractional complex transformation of the concern fractional differential equation to convert it for the solvable integer order differential equation. After the successful implementation of the presented methods, a comprehensive class of novel and broad-ranging exact and solitary travelling wave solutions were discovered, in terms of trigonometric, rational and hyperbolic functions. Hence, the present methods are reliable and efficient for solving nonlinear fractional problems in mathematics physics. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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24 pages, 444 KiB

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Interior Elastic Scattering by a Non-Penetrable Partially Coated Obstacle and Its Shape Recovering

by Angeliki Kaiafa and Vassilios Sevroglou

Mathematics 2021, 9(19), 2485; https://doi.org/10.3390/math9192485 - 04 Oct 2021

Cited by 1 | Viewed by 1320

Abstract

In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due [...] Read more.

In this paper, the interior elastic direct and inverse scattering problem of time-harmonic waves for a non-penetrable partially coated obstacle placed in a homogeneous and isotropic medium is studied. The scattering problem is formulated via the Navier equation, considering incident circular waves due to point-source fields, where the corresponding scattered data are measured on a closed curve inside the obstacle. Our model, from the mathematical point of view, is described by a mixed boundary value problem in which the scattered field satisfies mixed Dirichlet-Robin boundary conditions on the Lipschitz boundary of the obstacle. Using a variational equation method in an appropriate Sobolev space setting, uniqueness and existence results as well as stability ones are established. The corresponding inverse problem is also studied, and using some specific auxiliary integral operators an appropriate modified factorisation method is given. In addition, an inversion algorithm for shape recovering of the partially coated obstacle is presented and proved. Last but not least, useful remarks and conclusions concerning the direct scattering problem and its linchpin with the corresponding inverse one are given. Full article

(This article belongs to the Special Issue Applications of Partial Differential Equations in Mathematical Physics, 1st Edition)

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