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Advances and Novel Analytical Approaches to Boundary Value Problems in Engineering Sciences

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

Deadline for manuscript submissions: closed (20 December 2023) | Viewed by 8596

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Special Issue Editors

Prof. Dr. Stanislav Olegovich Papkov

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Guest Editor

Department of Mathematics, Sevastopol State University, 299053 Sevastopol, Russia
Interests: dynamics and vibrations: vibrations of plates; wave propagation; vibration and stability; composite plates; dynamic stiffness method; analytical methods of solution boundary problems; superposition method and its different modifications; theory of infinite systems of linear equations; theory of integral equations; acoustic; underwater wave propagation; theory of plane-layered waveguides

Prof. Dr. Lizhong Jiang

E-Mail Website
Guest Editor

School of Civil Engineering, Central South University, Changsha 410083, China
Interests: analytical modelling of bridges; structural dynamics; structural instabilities; seismic response of bridges; anti-seismic and stability design of structures

Prof. Dr. Xiang Liu

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Guest Editor

Key Laboratory of Traffic Safety on Track, Ministry of Education, School of Traffic and Transportation Engineering, Central South University, Changsha 410075, China
Interests: mathematical modelling; eigenvalue problems; analytical methods; uncertainty qualification; noise and vibration structure; dynamics fluid structure interaction; structure instabilities; computational mechanics

Special Issue Information

Dear Colleagues,

We invite you to submit your latest research in the area of analytical and semi-analytical approaches to the solutions of boundary value problems in various fields of engineering sciences. It is well-acknowledged that it is an important but challenging task to develop exact and/or analytical solutions for engineering problems. The present Special Issue of Mathematics emphasizes novel analytical and semi-analytical approaches and recent developments in the investigation of boundary value problems of partial differential equations arising from engineering sciences. We also welcome the submission of research papers investigating technical models with singularities. Potential topics include, but are not limited to, mechanical engineering, acoustics, mathematical physics, and applied mathematics.

Prof. Dr. Stanislav Olegovich Papkov
Prof. Dr. Lizhong Jiang
Prof. Dr. Xiang Liu
Guest Editors

Manuscript Submission Information

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

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

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

Keywords

boundary value problems
analytical approaches
singularities
acoustics
aerospace
dynamics
engineering
elasticity
modeling

Published Papers (6 papers)

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Research

18 pages, 648 KiB

Open AccessFeature PaperArticle

Weak and Classical Solutions to Multispecies Advection–Dispersion Equations in Multilayer Porous Media

by Miglena N. Koleva and Lubin G. Vulkov

Mathematics 2023, 11(14), 3103; https://doi.org/10.3390/math11143103 - 13 Jul 2023

Cited by 2 | Viewed by 752

Abstract

The basic model motivating this work is that of contaminant transport in the Earth’s subsurface, which contains layers in which analytical and semi-analytical solutions of the corresponding advection–dispersion equations could be derived. Then, using the interface relations between adjacent layers, one can streamline the study of the model to the solution to the initial boundary value problem for a coupled parabolic system on partitioned domains. For IBVPs, we set up weak formulations and prove the existence and uniqueness of solutions to appropriate Sobolev-like spaces. A priori estimates at different levels of input data smoothness were obtained. The nonnegativity preservation over time of the solution is discussed. We numerically demonstrate how to solve the reduced truncated problem instead of the original multispecies one with a large number of layers. Full article

(This article belongs to the Special Issue Advances and Novel Analytical Approaches to Boundary Value Problems in Engineering Sciences)

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10 pages, 244 KiB

Open AccessArticle

On Kirchhoff-Type Equations with Hardy Potential and Berestycki–Lions Conditions

by Hua Yang and Jiu Liu

Mathematics 2023, 11(12), 2648; https://doi.org/10.3390/math11122648 - 9 Jun 2023

Viewed by 704

Abstract

The purpose of this paper is to investigate the existence and asymptotic properties of solutions to a Kirchhoff-type equation with Hardy potential and Berestycki–Lions conditions. Firstly, we show that the equation has a positive radial ground-state solution

u_{λ}

by using the Pohozaev [...] Read more.

u_{λ}

by using the Pohozaev manifold. Secondly, we prove that the solution

u_{λ_{n}}

, up to a subsequence, converges to a radial ground-state solution of the corresponding limiting equations as

λ_{n} \to 0^{-}

. Finally, we provide a brief summary. Full article

(This article belongs to the Special Issue Advances and Novel Analytical Approaches to Boundary Value Problems in Engineering Sciences)

15 pages, 294 KiB

Open AccessArticle

Dirichlet and Neumann Boundary Value Problems for Dunkl Polyharmonic Equations

by Hongfen Yuan and Valery Karachik

Mathematics 2023, 11(9), 2185; https://doi.org/10.3390/math11092185 - 5 May 2023

Viewed by 963

Abstract

Dunkl operators are a family of commuting differential–difference operators associated with a finite reflection group. These operators play a key role in the area of harmonic analysis and theory of spherical functions. We study the solution of the inhomogeneous Dunkl polyharmonic equation based on the solutions of Dunkl–Possion equations. Furthermore, we construct the solutions of Dirichlet and Neumann boundary value problems for Dunkl polyharmonic equations without invoking the Green’s function. Full article

(This article belongs to the Special Issue Advances and Novel Analytical Approaches to Boundary Value Problems in Engineering Sciences)

14 pages, 1842 KiB

Open AccessArticle

A New Method for Free Vibration Analysis of Triangular Isotropic and Orthotropic Plates of Isosceles Type Using an Accurate Series Solution

by Stanislav Papkov and Jnan Ranjan Banerjee

Mathematics 2023, 11(3), 649; https://doi.org/10.3390/math11030649 - 27 Jan 2023

Cited by 1 | Viewed by 1213

Abstract

In this paper, a new method based on an accurate analytical series solution for free vibration of triangular isotropic and orthotropic plates is presented. The proposed solution is expressed in terms of undetermined arbitrary coefficients, which are exactly satisfied by the governing differential equation in free vibration. The approach used is based on an innovative extension of the superposition method through the application of a modified system of trigonometric functions. The boundary conditions for bending displacements and bending rotations on the sides of the triangular plate led to an infinite system of linear algebraic equations in terms of the undetermined coefficients. Following this development, the paper then presents an algorithm to solve the boundary value problem for isotropic and orthotropic triangular plates for any kinematic boundary conditions. Of course, the boundary conditions with zero displacements and zero rotations on all sides correspond to the case when the plate is fully clamped all around. The convergence of the proposed method is examined by numerical simulation applying stringent accuracy requirements to fulfill the prescribed boundary conditions. Some of the computed numerical results are compared with published results and finally, the paper draws significant conclusions. Full article

(This article belongs to the Special Issue Advances and Novel Analytical Approaches to Boundary Value Problems in Engineering Sciences)

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19 pages, 300 KiB

Open AccessArticle

Existence of Positive Solutions for a Fully Fourth-Order Boundary Value Problem

by Yongxiang Li and Weifeng Ma

Mathematics 2022, 10(17), 3063; https://doi.org/10.3390/math10173063 - 25 Aug 2022

Cited by 3 | Viewed by 933

Abstract

This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem

u^{(4)} = f (t, u, u^{'}, u^{″}, u^{‴})

[0, 1]

with [...] Read more.

This paper deals with the existence of positive solutions of the fully fourth-order boundary value pqroblem

u^{(4)} = f (t, u, u^{'}, u^{″}, u^{‴})

[0, 1]

with the boundary condition

u (0) = u (1) = u^{″} (0) = u^{″} (1) = 0

, which models a statically bending elastic beam whose two ends are simply supported, where

f : [0, 1] \times R^{+} \times R \times R^{-} \times R \to R^{+}

is continuous. Some precise inequality conditions on f guaranteeing the existence of positive solutions are presented. The inequality conditions allow that

f (t, u, v, w, z)

may be asymptotically linear, superlinear, or sublinear on

u, v, w

, and z as

| (u, v, w, z) | \to 0

and

| (u, v, w, z) | \to \infty

. Our discussion is based on the fixed point index theory in cones. Full article

(This article belongs to the Special Issue Advances and Novel Analytical Approaches to Boundary Value Problems in Engineering Sciences)

23 pages, 2068 KiB

Open AccessArticle

Extension of the Wittrick-Williams Algorithm for Free Vibration Analysis of Hybrid Dynamic Stiffness Models Connecting Line and Point Nodes

by Xiang Liu, Shaoqi Qiu, Suchao Xie and Jnan Ranjan Banerjee

Mathematics 2022, 10(1), 57; https://doi.org/10.3390/math10010057 - 24 Dec 2021

Cited by 6 | Viewed by 2988

Abstract

This paper extends the Wittrick-Williams (W-W) algorithm for hybrid dynamic stiffness (DS) models connecting any combinations of line and point nodes. The principal novelties lie in the development of both the DS formulation and the solution technique in a sufficiently systematic and general manner. The parent structure is considered to be in the form of two dimensional DS elements with line nodes, which can be connected to rigid/spring point supports/connections, rod/beam point supports/connections, and point connections to substructures. This is achieved by proposing a direct constrain method in a strong form which makes the modeling process straightforward. For the solution technique, the W-W algorithm is extended for all of the above hybrid DS models. No matrix inversion is needed in the proposed extension, making the algorithm numerically stable, especially for complex built-up structures. A mathematical proof is provided for the extended W-W algorithm. The proposed DS formulation and the extended W-W algorithm are validated by the FE results computed by ANSYS. This work significantly extends the application scope of the DS formulation and the W-W algorithm in a methodical and reliable manner, providing a powerful eigenvalue analysis tool for beam-plate built-up structures. Full article

(This article belongs to the Special Issue Advances and Novel Analytical Approaches to Boundary Value Problems in Engineering Sciences)

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