Mathematics

Research

20 pages, 4135 KiB

Open AccessArticle

Estimation of the Total Heat Exchange Factor for the Reheating Furnace Based on the First-Optimize-Then-Discretize Approach and an Improved Hybrid Conjugate Gradient Algorithm

by Zhi Yang, Xiaochuan Luo, Pengbo Liu, Jinwei Qiao and Ming Liu

Mathematics 2022, 10(21), 4074; https://doi.org/10.3390/math10214074 - 2 Nov 2022

Viewed by 1110

Abstract

The total heat exchange factor is one of the most important thermal physical parameters in the heat transfer model for a reheating furnace machine. In this paper, a novel general strategy, which is combined with the first-optimize-then-discretize (FOTD) approach and an improved hybrid [...] Read more.

The total heat exchange factor is one of the most important thermal physical parameters in the heat transfer model for a reheating furnace machine. In this paper, a novel general strategy, which is combined with the first-optimize-then-discretize (FOTD) approach and an improved hybrid conjugate gradient (IHCG) algorithm, is proposed to identify the total heat exchange factor by solving a nonlinear inverse heat conduction problem (IHCP). Firstly, a nonlinear IHCP with the Dirichlet-type boundary condition

T_{m} (t) = T (0, t)

is built to determine the unknown total heat exchange factor

w (t)

. Secondly, the analysis of the Fréchet gradient of the cost functional is given and the gradient is proved as Lipschitz continuous by the FOTD approach. Thirdly, based on the gradient information by FOTD, a new IHCG algorithm, whose global convergence is proved by us, is proposed for fast solving of the optimization problem. Finally, simulation experiments are given to verify the effectiveness of the proposed strategy. Compared with the first-discretize-then-optimize (FDTO) approach, the FOTD approach can reduce running time and iteration number. Compared with other CG algorithms, the proposed IHCG algorithm has better convergence performance. The experimental data by the thermocouples experiments from a reheating furnace are also given to identify the total heat exchange factor. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)

► Show Figures

Figure 1

31 pages, 490 KiB

Open AccessArticle

Algebraic Construction of the Sigma Function for General Weierstrass Curves

by Jiryo Komeda, Shigeki Matsutani and Emma Previato

Mathematics 2022, 10(16), 3010; https://doi.org/10.3390/math10163010 - 20 Aug 2022

Cited by 3 | Viewed by 1262 | Correction

Abstract

The Weierstrass curve X is a smooth algebraic curve determined by the Weierstrass canonical form, [...] Read more.

The Weierstrass curve X is a smooth algebraic curve determined by the Weierstrass canonical form,

y^{r} + A_{1} (x) y^{r - 1} + A_{2} (x) y^{r - 2} + \dots + A_{r - 1} (x) y + A_{r} (x) = 0

, where r is a positive integer, and each

A_{j}

is a polynomial in x with a certain degree. It is known that every compact Riemann surface has a Weierstrass curve X, which is birational to the surface. The form provides the projection

ϖ_{r} : X \to P

as a covering space. Let

R_{X} : = H^{0} (X, O_{X} (* \infty))

and

R_{P} : = H^{0} (P, O_{P} (* \infty))

. Recently, we obtained the explicit description of the complementary module

R_{X}^{c}

of

R_{P}

-module

R_{X}

, which leads to explicit expressions of the holomorphic form except ∞,

H^{0} (P, A_{P} (* \infty))

and the trace operator

p_{X}

such that

p_{X} (P, Q) = δ_{P, Q}

for

ϖ_{r} (P) = ϖ_{r} (Q)

for

P, Q \in X \ {\infty}

. In terms of these, we express the fundamental two-form of the second kind

Ω

and its connection to the sigma function for X. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)

20 pages, 5140 KiB

Open AccessArticle

Conjugate Natural Convection of a Hybrid Nanofluid in a Cavity Filled with Porous and Non-Newtonian Layers: The Impact of the Power Law Index

by Mohamed Omri, Muhammad Jamal, Shafqat Hussain, Lioua Kolsi and Chemseddine Maatki

Mathematics 2022, 10(12), 2044; https://doi.org/10.3390/math10122044 - 13 Jun 2022

Cited by 7 | Viewed by 1443

Abstract

This study deals with the effect of the power law index on the convective heat transfer of hybrid nanofluids in a square cavity divided into three layers. The effect of a solid fluid layer is also given attention. A two-dimensional system of partial [...] Read more.

This study deals with the effect of the power law index on the convective heat transfer of hybrid nanofluids in a square cavity divided into three layers. The effect of a solid fluid layer is also given attention. A two-dimensional system of partial differential equations is discretized by using the generalized finite element method (FEM). A FEM having cubic polynomials (P3) is employed to approximate the temperature and velocity components, whereas the pressure is approached using quadratic finite element functions. The discretized set of equations have been solved using Newton’s method. The numerical code which is used in this study has been validated by comparing with experimental findings. Mathematical simulations are performed for different sets of parameters, including the Rayleigh number (between

10^{3}

and

10^{6}

), the power law index (between

0.6

to

1.8

), Darcy number (between

10^{- 6}

to

10^{- 2}

), undulation (between 1 and 5) and the thermal conductivity ratio (between

0.1

and 10). The results infer that a remarkable penetration of streamlines is figured out towards the porous hybrid layer as the power law index is increased. The average

N u

increases with increasing

R a

, and the maximum value is noted at

R a = 10^{6}

. There is no much alteration observed for isotherms at the solid layer by increasing

D a

. The average

N u

decreases by increasing the undulations. The rate of heat transfer is enhanced at the heated boundary and solid fluid interface of the cavity by raising the ratio of thermal conductivity. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)

► Show Figures

Figure 1

21 pages, 359 KiB

Open AccessArticle

Analytic Expressions for Debye Functions and the Heat Capacity of a Solid

by Ivan Gonzalez, Igor Kondrashuk, Victor H. Moll and Alfredo Vega

Mathematics 2022, 10(10), 1745; https://doi.org/10.3390/math10101745 - 20 May 2022

Cited by 2 | Viewed by 1864

Abstract

Analytic expressions for the N-dimensional Debye function are obtained by the method of brackets. The new expressions are suitable for the study of heat capacity of solids and the analysis of the asymptotic behavior of this function, both in the high and [...] Read more.

Analytic expressions for the N-dimensional Debye function are obtained by the method of brackets. The new expressions are suitable for the study of heat capacity of solids and the analysis of the asymptotic behavior of this function, both in the high and low temperature limits. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)

13 pages, 301 KiB

Open AccessArticle

Estimates of Mild Solutions of Navier–Stokes Equations in Weak Herz-Type Besov–Morrey Spaces

by Ruslan Abdulkadirov and Pavel Lyakhov

Mathematics 2022, 10(5), 680; https://doi.org/10.3390/math10050680 - 22 Feb 2022

Cited by 4 | Viewed by 3476

Abstract

The main goal of this article is to provide estimates of mild solutions of Navier–Stokes equations with arbitrary external forces in

R^{n}

for

n \geq 2

on proposed weak Herz-type Besov–Morrey spaces. These spaces are larger than known Besov–Morrey and Herz spaces [...] Read more.

The main goal of this article is to provide estimates of mild solutions of Navier–Stokes equations with arbitrary external forces in

R^{n}

for

n \geq 2

on proposed weak Herz-type Besov–Morrey spaces. These spaces are larger than known Besov–Morrey and Herz spaces considered in known works on Navier–Stokes equations. Morrey–Sobolev and Besov–Morrey spaces based on weak-Herz space denoted as

W {\dot{K}}_{p, q}^{α} M_{μ}^{s}

and

W {\dot{K}}_{p, q}^{α} {\dot{N}}_{μ, r}^{s}

, respectively, represent new properties and interpolations. This class of spaces and its developed properties could also be employed to study elliptic, parabolic, and conservation-law type PDEs. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)

18 pages, 825 KiB

Open AccessArticle

Structural Stability on the Boundary Coefficient of the Thermoelastic Equations of Type III

by Xuejiao Chen and Yuanfei Li

Mathematics 2022, 10(3), 366; https://doi.org/10.3390/math10030366 - 25 Jan 2022

Cited by 4 | Viewed by 1661

Abstract

This paper investigates the spatial behavior of the solutions of thermoelastic equations of type III in a semi-infinite cylinder by using the partial differential inequalities. By setting an arbitrary positive constant in the energy expression, the fast decay rate of the solutions is [...] Read more.

This paper investigates the spatial behavior of the solutions of thermoelastic equations of type III in a semi-infinite cylinder by using the partial differential inequalities. By setting an arbitrary positive constant in the energy expression, the fast decay rate of the solutions is obtained. Based on the results of decay, the continuous dependence and the convergence results on the boundary coefficient are established by using the differential inequality technique and the energy analysis method. The main work of this paper is to extend the study of continuous dependence to a semi-infinite cylinder, which can be used as a reference for the study of other types of partial differential equations. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)

18 pages, 330 KiB

Open AccessArticle

Characterizing Base in Warped Product Submanifolds of Complex Projective Spaces by Differential Equations

by Ali H. Alkhaldi, Pişcoran Laurian-Ioan, Izhar Ahmad and Akram Ali

Mathematics 2022, 10(2), 244; https://doi.org/10.3390/math10020244 - 13 Jan 2022

Viewed by 1044

Abstract

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold

M^{n}

in a complex projective space is presented. Some characterizations of the base [...] Read more.

In this study, a link between the squared norm of the second fundamental form and the Laplacian of the warping function for a warped product pointwise semi-slant submanifold

M^{n}

in a complex projective space is presented. Some characterizations of the base

N_{T}

of

M^{n}

are offered as applications. We also look at whether the base

N_{T}

is isometric to the Euclidean space

R^{p}

or the Euclidean sphere

S^{p}

, subject to some constraints on the second fundamental form and warping function. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)

14 pages, 286 KiB

Open AccessArticle

Heterogeneous Diffusion, Stability Analysis, and Solution Profiles for a MHD Darcy–Forchheimer Model

by José Luis Díaz, Saeed Rahman and Juan Miguel García-Haro

Mathematics 2022, 10(1), 20; https://doi.org/10.3390/math10010020 - 21 Dec 2021

Cited by 3 | Viewed by 2087

Abstract

In the presented analysis, a heterogeneous diffusion is introduced to a magnetohydrodynamics (MHD) Darcy–Forchheimer flow, leading to an extended Darcy–Forchheimer model. The introduction of a generalized diffusion was proposed by Cohen and Murray to study the energy gradients in spatial structures. In addition, [...] Read more.

In the presented analysis, a heterogeneous diffusion is introduced to a magnetohydrodynamics (MHD) Darcy–Forchheimer flow, leading to an extended Darcy–Forchheimer model. The introduction of a generalized diffusion was proposed by Cohen and Murray to study the energy gradients in spatial structures. In addition, Peletier and Troy, on one side, and Rottschäfer and Doelman, on the other side, have introduced a general diffusion (of a fourth-order spatial derivative) to study the oscillatory patterns close the critical points induced by the reaction term. In the presented study, analytical conceptions to a proposed problem with heterogeneous diffusions are introduced. First, the existence and uniqueness of solutions are provided. Afterwards, a stability study is presented aiming to characterize the asymptotic convergent condition for oscillatory patterns. Dedicated solution profiles are explored, making use of a Hamilton–Jacobi type of equation. The existence of oscillatory patterns may induce solutions to be negative, close to the null equilibrium; hence, a precise inner region of positive solutions is obtained. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)

Review

Jump to: Research, Other

20 pages, 991 KiB

Open AccessReview

Hyperelliptic Functions and Motion in General Relativity

by Saskia Grunau and Jutta Kunz

Mathematics 2022, 10(12), 1958; https://doi.org/10.3390/math10121958 - 7 Jun 2022

Viewed by 1581

Abstract

Analysis of black hole spacetimes requires study of the motion of particles and light in these spacetimes. Here exact solutions of the geodesic equations are the means of choice. Numerous interesting black hole spacetimes have been analyzed in terms of elliptic functions. However, [...] Read more.

Analysis of black hole spacetimes requires study of the motion of particles and light in these spacetimes. Here exact solutions of the geodesic equations are the means of choice. Numerous interesting black hole spacetimes have been analyzed in terms of elliptic functions. However, the presence of a cosmological constant, higher dimensions or alternative gravity theories often necessitate an analysis in terms of hyperelliptic functions. Here we review the method and current status for solving the geodesic equations for the general hyperelliptic case, illustrating it with a set of examples of genus

g = 2

: higher dimensional Schwarzschild black holes, rotating dyonic

U {(1)}^{2}

black holes, and black rings. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)

► Show Figures

Figure 1

19 pages, 368 KiB

Open AccessReview

Partial Differential Equations and Quantum States in Curved Spacetimes

by Zhirayr Avetisyan and Matteo Capoferri

Mathematics 2021, 9(16), 1936; https://doi.org/10.3390/math9161936 - 13 Aug 2021

Cited by 2 | Viewed by 2081

Abstract

In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction [...] Read more.

In this review paper, we discuss the relation between recent advances in the theory of partial differential equations and their applications to quantum field theory on curved spacetimes. In particular, we focus on hyperbolic propagators and the role they play in the construction of physically admissible quantum states—the so-called Hadamard states—on globally hyperbolic spacetimes. We will review the notion of a propagator and discuss how it can be constructed in an explicit and invariant fashion, first on a Riemannian manifold and then on a Lorentzian spacetime. Finally, we will recall the notion of Hadamard state and relate the latter to hyperbolic propagators via the wavefront set, a subset of the cotangent bundle capturing the information about the singularities of a distribution. Full article

(This article belongs to the Special Issue Partial Differential Equations and Applications)