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Mathematics
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Recent Developments in Boundary Value Problems: Theoretical and Computational Aspects
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Recent Developments in Boundary Value Problems: Theoretical and Computational Aspects

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Mathematics and Computer Science".

Deadline for manuscript submissions: closed (15 October 2020) | Viewed by 9880

Special Issue Editors

Prof. Dr. Farzana A. McRae

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Guest Editor
Department of Mathematics, Catholic University of America, Washington, DC 20064, USA
Interests: fractional differential equations; boundary value problems
Prof. Dr. Zahia Drici

E-Mail Website
Guest Editor
Department of Mathematics, Illinois Wesleyan University, P.O. Box 2900, Bloomington, IL, USA
Interests: ordinary differential equations; dynamical systems; difference and functional equations; integral equations; electromagnetic theory and its applications
Prof. Dr. J. Vasundhara Devi

E-Mail Website
Guest Editor
Lakshmikantham Institute for Advanced Studies, Gayatri Vidya Parishad College of Engineering (Autonomous),Viskhapatnam, India
Interests: differential equations; nonlinear analysis

Special Issue Information

Dear Colleagues,

Boundary-value problems arise in a natural way in many applied fields and fall into different categories, such as ordinary, partial, integrodifferential, functional, impulsive, inverse, and fractional boundary value problems, and also according to the type of boundary conditions, such as two-point, periodic, multipoint, local and nonlocal, integral, and homogeneous and non-homogeneous boundary value conditions. Research in boundary value problems addresses theoretical aspects, such as the existence and uniqueness of solutions, as well as computational aspects, such as methods for approximating solutions.

This Special Issue will focus on recent theoretical and computational developments of boundary value problems and their applications. Topics include but are not limited to (i) the existence and uniqueness of solutions, (ii) analytical methods, (iii) iterative methods, (iv) perturbation techniques, (v) numerical techniques, (vi) applications in physical, social and life sciences, (vii) applications in engineering, and (viii) applications in economics and social sciences.

Prof. Dr. Farzana A. McRae
Prof. Dr. Zahia Drici
Prof. Dr. J. Vasundhara Devi
Guest Editors

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Published Papers (4 papers)

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Research

19 pages, 309 KiB  
Article
Efficient Numerical Scheme for the Solution of Tenth Order Boundary Value Problems by the Haar Wavelet Method
by Rohul Amin, Kamal Shah, Imran Khan, Muhammad Asif, Mehdi Salimi and Ali Ahmadian
Mathematics 2020, 8(11), 1874; https://doi.org/10.3390/math8111874 - 29 Oct 2020
Cited by 8 | Viewed by 2725
Abstract
In this paper, an accurate and fast algorithm is developed for the solution of tenth order boundary value problems. The Haar wavelet collocation method is applied to both linear and nonlinear boundary value problems. In this technqiue, the tenth order derivative in boundary [...] Read more.
In this paper, an accurate and fast algorithm is developed for the solution of tenth order boundary value problems. The Haar wavelet collocation method is applied to both linear and nonlinear boundary value problems. In this technqiue, the tenth order derivative in boundary value problem is approximated using Haar functions and the process of integration is used to obtain the expression of lower order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking validation and the convergence of the proposed technique. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The experimental rate of convergence using different number of collocation points is also calculated, which is nearly equal to 2. Full article
(This article belongs to the Special Issue Recent Developments in Boundary Value Problems: Theoretical and Computational Aspects)
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23 pages, 5441 KiB  
Article
Space–Time Radial Basis Function–Based Meshless Approach for Solving Convection–Diffusion Equations
by Cheng-Yu Ku, Jing-En Xiao and Chih-Yu Liu
Mathematics 2020, 8(10), 1735; https://doi.org/10.3390/math8101735 - 10 Oct 2020
Cited by 4 | Viewed by 2435
Abstract
This article proposes a space–time meshless approach based on the transient radial polynomial series function (TRPSF) for solving convection–diffusion equations. We adopted the TRPSF as the basis function for the spatial and temporal discretization of the convection–diffusion equation. The TRPSF is constructed in [...] Read more.
This article proposes a space–time meshless approach based on the transient radial polynomial series function (TRPSF) for solving convection–diffusion equations. We adopted the TRPSF as the basis function for the spatial and temporal discretization of the convection–diffusion equation. The TRPSF is constructed in the space–time domain, which is a combination of n–dimensional Euclidean space and time into an n + 1–dimensional manifold. Because the initial and boundary conditions were applied on the space–time domain boundaries, we converted the transient problem into an inverse boundary value problem. Additionally, all partial derivatives of the proposed TRPSF are a series of continuous functions, which are nonsingular and smooth. Solutions were approximated by solving the system of simultaneous equations formulated from the boundary, source, and internal collocation points. Numerical examples including stationary and nonstationary convection–diffusion problems were employed. The numerical solutions revealed that the proposed space–time meshless approach may achieve more accurate numerical solutions than those obtained by using the conventional radial basis function (RBF) with the time-marching scheme. Furthermore, the numerical examples indicated that the TRPSF is more stable and accurate than other RBFs for solving the convection–diffusion equation. Full article
(This article belongs to the Special Issue Recent Developments in Boundary Value Problems: Theoretical and Computational Aspects)
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15 pages, 1735 KiB  
Article
A Revisit of the Boundary Value Problem for Föppl–Hencky Membranes: Improvement of Geometric Equations
by Yong-Sheng Lian, Jun-Yi Sun, Zhi-Hang Zhao, Xiao-Ting He and Zhou-Lian Zheng
Mathematics 2020, 8(4), 631; https://doi.org/10.3390/math8040631 - 20 Apr 2020
Cited by 19 | Viewed by 2390
Abstract
In this paper, the well-known Föppl–Hencky membrane problem—that is, the problem of axisymmetric deformation of a transversely uniformly loaded and peripherally fixed circular membrane—was resolved, and a more refined closed-form solution of the problem was presented, where the so-called small rotation angle assumption [...] Read more.
In this paper, the well-known Föppl–Hencky membrane problem—that is, the problem of axisymmetric deformation of a transversely uniformly loaded and peripherally fixed circular membrane—was resolved, and a more refined closed-form solution of the problem was presented, where the so-called small rotation angle assumption of the membrane was given up. In particular, a more effective geometric equation was, for the first time, established to replace the classic one, and finally the resulting new boundary value problem due to the improvement of geometric equation was successfully solved by the power series method. The conducted numerical example indicates that the closed-form solution presented in this study has higher computational accuracy in comparison with the existing solutions of the well-known Föppl–Hencky membrane problem. In addition, some important issues were discussed, such as the difference between membrane problems and thin plate problems, reasonable approximation or assumption during establishing geometric equations, and the contribution of reducing approximations or relaxing assumptions to the improvement of the computational accuracy and applicability of a solution. Finally, some opinions on the follow-up work for the well-known Föppl–Hencky membrane were presented. Full article
(This article belongs to the Special Issue Recent Developments in Boundary Value Problems: Theoretical and Computational Aspects)
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19 pages, 293 KiB  
Article
On the Solvability of Fourth-Order Two-Point Boundary Value Problems
by Ravi P. Agarwal and Petio S. Kelevedjiev
Mathematics 2020, 8(4), 603; https://doi.org/10.3390/math8040603 - 16 Apr 2020
Cited by 5 | Viewed by 1763
Abstract
In this paper, we study the solvability of various two-point boundary value problems for [...] Read more.
In this paper, we study the solvability of various two-point boundary value problems for x ( 4 ) = f ( t , x , x ′ , x ″ , x ‴ ) , t ∈ ( 0 , 1 ) , where f may be defined and continuous on a suitable bounded subset of its domain. Imposing barrier strips type conditions, we give results guaranteeing not only positive solutions, but also monotonic ones and such with suitable curvature. Full article
(This article belongs to the Special Issue Recent Developments in Boundary Value Problems: Theoretical and Computational Aspects)
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