scholarly journals Sturm–Liouville Differential Equations Involving Kurzweil–Henstock Integrable Functions

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1403
Author(s):  
Salvador Sánchez-Perales ◽  
Tomás Pérez-Becerra ◽  
Virgilio Vázquez-Hipólito ◽  
José J. Oliveros-Oliveros
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Existence And Uniqueness ◽  
Dirichlet Boundary ◽  
Sufficient Conditions ◽  
Integrable Functions ◽  
Uniqueness Of The Solution ◽  
Sturm Liouville ◽  
Unbounded Intervals

In this paper, we give sufficient conditions for the existence and uniqueness of the solution of Sturm–Liouville equations subject to Dirichlet boundary value conditions and involving Kurzweil–Henstock integrable functions on unbounded intervals. We also present a finite element method scheme for Kurzweil–Henstock integrable functions.

scholarly journals Conditions to Guarantee the Existence of the Solution to Stochastic Differential Equations of Neutral Type

Symmetry ◽  
2020 ◽  
Vol 12 (10) ◽  
pp. 1613
Author(s):  
Mun-Jin Bae ◽  
Chan-Ho Park ◽  
Young-Ho Kim
Keyword(s):  
Differential Equations ◽  
Stochastic Differential Equations ◽  
Existence And Uniqueness ◽  
Neutral Type ◽  
Sufficient Conditions ◽  
Exponential Estimates ◽  
Analysis Theory ◽  
Linear Growth Condition ◽  
Uniqueness Of The Solution ◽  
The Uniqueness Theorem

The main purpose of this study was to demonstrate the existence and the uniqueness theorem of the solution of the neutral stochastic differential equations under sufficient conditions. As an alternative to the stochastic analysis theory of the neutral stochastic differential equations, we impose a weakened Ho¨lder condition and a weakened linear growth condition. Stochastic results are obtained for the theory of the existence and uniqueness of the solution. We first show that the conditions guarantee the existence and uniqueness; then, we show some exponential estimates for the solutions.

scholarly journals CONVECTIVE AND RADIATIVE THERMAL TRANSFER WITH MULTIPLE REFLECTIONS: ANALYSIS AND APPROXIMATION BY A FINITE ELEMENT METHOD

2001 ◽  
Vol 11 (02) ◽  
pp. 229-262 ◽  
Author(s):  
J. MONNIER ◽  
J. P. VILA
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Existence And Uniqueness ◽  
Multiple Reflections ◽  
Convection Diffusion ◽  
Thermal Transfer ◽  
Uniqueness Of The Solution ◽  
Nonlinear Integrodifferential System ◽  
Account Heat ◽  
Element Method

We study a 3D steady-state thermal model taking into account heat transfer by convection, diffusion and radiation with multiple reflections (grey bodies). This model is a nonlinear integrodifferential system which we solve numerically by a finite element method. Some results of existence and uniqueness of the solution are proved, the numerical analysis is detailed, error estimates are given and two-dimensional numerical results of thermal exchanges under a car bonnet are presented.

scholarly journals Study of a nonlinear multi-terms boundary value problem of fractional pantograph differential equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Muhammad Bahar Ali Khan ◽  
Thabet Abdeljawad ◽  
Kamal Shah ◽  
Gohar Ali ◽  
Hasib Khan ◽  
...  
Keyword(s):  
Differential Equations ◽  
Delay Differential Equations ◽  
Existence And Uniqueness ◽  
Fixed Point Theorems ◽  
Research Work ◽  
Sufficient Conditions ◽  
Point Of View ◽  
Uniqueness Of The Solution ◽  
Delay Differential ◽  
Ulam Type Stability

AbstractIn this research work, a class of multi-term fractional pantograph differential equations (FODEs) subject to antiperiodic boundary conditions (APBCs) is considered. The ensuing problem involves proportional type delay terms and constitutes a subclass of delay differential equations known as pantograph. On using fixed point theorems due to Banach and Schaefer, some sufficient conditions are developed for the existence and uniqueness of the solution to the problem under investigation. Furthermore, due to the significance of stability analysis from a numerical and optimization point of view Ulam type stability and its various forms are studied. Here we mention different forms of stability: Hyers–Ulam (HU), generalized Hyers–Ulam (GHU), Hyers–Ulam Rassias (HUR) and generalized Hyers–Ulam–Rassias (GHUR). After the demonstration of our results, some pertinent examples are given.

scholarly journals The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Huanting Li ◽  
Yunfei Peng ◽  
Kuilin Wu
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Continuous Dependence ◽  
Nonlinear Differential Equations ◽  
Sufficient Conditions ◽  
Initial State ◽  
Switching Line ◽  
Uniqueness Of The Solution ◽  
The Stability ◽  
Necessary And Sufficient

<p style='text-indent:20px;'>In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching <inline-formula><tex-math id="M1">\begin{document}$ k\in \mathbb{N} $\end{document}</tex-math></inline-formula> times on bounded time intervals.</p>

scholarly journals Numerical Solution of Some Differential Equations with Henstock–Kurzweil Functions

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
D. A. León-Velasco ◽  
M. M. Morín-Castillo ◽  
J. J. Oliveros-Oliveros ◽  
T. Pérez-Becerra ◽  
J. A. Escamilla-Reyna
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Differential Equations ◽  
Numerical Solution ◽  
Elliptic Problem ◽  
The Finite Element Method ◽  
Weak Formulation ◽  
Numerical Examples ◽  
Integrable Functions ◽  
Bounded Variation Functions

In this work, the Finite Element Method is used for finding the numerical solution of an elliptic problem with Henstock–Kurzweil integrable functions. In particular, Henstock–Kurzweil high oscillatory functions were considered. The weak formulation of the problem leads to integrals that are calculated using some special quadratures. Definitions and theorems were used to guarantee the existence of the integrals that appear in the weak formulation. This allowed us to apply the above formulation for the type of slope bounded variation functions. Numerical examples were developed to illustrate the ideas presented in this article.

The Characteristic Radial Basis Meshless Method for Convection-Dominated Diffusion Equations

2012 ◽  
Vol 182-183 ◽  
pp. 1756-1760 ◽  
Author(s):  
Xin Qiang Qin ◽  
Xian Bao Duan ◽  
Wei Guo
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Meshless Method ◽  
Existence And Uniqueness ◽  
Diffusion Equations ◽  
Characteristic Finite Element ◽  
Radial Basis ◽  
Uniqueness Of The Solution ◽  
Convection Dominated ◽  
Diffusion Problems

A characteristic radial basis meshless method (CRBM) is developed for numerically solving convection-dominated diffusion equations. This method is a truly meshless technique without mesh discretization, and it is numerically stable and more efficient than the characteristic finite element method (CFEM) as demonstrated by the provided numerical results for convection-dominated diffusion problems. Moreover, the existence and uniqueness of the solution to the method are proved.

scholarly journals Finite Element Method for Linear Multiterm Fractional Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-9 ◽  
Author(s):  
Abdallah A. Badr
Keyword(s):  
Differential Equation ◽  
Finite Element Method ◽  
Finite Element ◽  
Numerical Methods ◽  
Existence And Uniqueness ◽  
The Finite Element Method ◽  
Galerkin Approach ◽  
Uniqueness Of The Solution ◽  
Fractional Differential ◽  
Element Method

We consider the linear multiterm fractional differential equation (fDE). Existence and uniqueness of the solution of such equation are discussed. We apply the finite element method (FEM) to obtain the numerical solution of this equation using Galerkin approach. A comparison, through examples, between our techniques and other previous numerical methods is established.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals On Existence Theorems to Symmetric Functional Set-Valued Differential Equations

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1219
Author(s):  
Marek T. Malinowski
Keyword(s):  
Differential Equations ◽  
Existence And Uniqueness ◽  
Existence Theorems ◽  
Type Theorem ◽  
Integral Representations ◽  
Uniqueness Of The Solution ◽  
Nonempty Compact ◽  
Picard Type Theorem ◽  
Convex Subsets

In this paper, we consider functional set-valued differential equations in their integral representations that possess integrals symmetrically on both sides of the equations. The solutions have values that are the nonempty compact and convex subsets. The main results contain a Peano type theorem on the existence of the solution and a Picard type theorem on the existence and uniqueness of the solution to such equations. The proofs are based on sequences of approximations that are constructed with appropriate Hukuhara differences of sets. An estimate of the magnitude of the solution’s values is provided as well. We show the closeness of the unique solutions when the equations differ slightly.

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