Weighted Estimates for Boundary Value Problems with Fractional Derivatives

2020 ◽  
Vol 20 (4) ◽  
pp. 609-630 ◽  
Author(s):  
Ivan P. Gavrilyuk ◽  
Volodymyr L. Makarov ◽  
Nataliya V. Mayko
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Dirichlet Boundary ◽  
Boundary Effect ◽  
Boundary Value ◽  
Grid Method ◽  
Weighted Estimates ◽  
Regularity Properties ◽  
Better Than

AbstractWe consider the Dirichlet boundary value problem for linear fractional differential equations with the Riemann–Liouville fractional derivatives. By transforming the boundary value problem to the integral equation, some regularity properties of the exact solution are derived. Based on these properties, the numerical solution of the boundary value problems by a grid method is discussed and weighted estimates considering the boundary effect are obtained. It is shown that the accuracy (the convergence rate) near the boundary is better than inside the domain due to the influence of the Dirichlet boundary condition.

scholarly journals Nonlocal Fractional Hybrid Boundary Value Problems Involving Mixed Fractional Derivatives and Integrals via a Generalization of Darbo’s Theorem

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Ayub Samadi ◽  
Sotiris K. Ntouyas ◽  
Jessada Tariboon
Keyword(s):  
Boundary Value Problem ◽  
Theoretical Result ◽  
Boundary Value Problems ◽  
Fractional Derivatives ◽  
Existence Result ◽  
Boundary Value ◽  
Measures Of Noncompactness ◽  
Fractional Derivatives And Integrals ◽  
Mixed Fractional Derivatives

In this work, a new existence result is established for a nonlocal hybrid boundary value problem which contains one left Caputo and one right Riemann–Liouville fractional derivatives and integrals. The main result is proved by applying a new generalization of Darbo’s theorem associated with measures of noncompactness. Finally, an example to justify the theoretical result is also presented.

scholarly journals Exact Multiplicity of Sign-Changing Solutions for a Class of Second-Order Dirichlet Boundary Value Problem with Weight Function

2013 ◽  
Vol 2013 ◽  
pp. 1-9
Author(s):  
Yulian An
Keyword(s):  
Boundary Value Problem ◽  
Weight Function ◽  
Boundary Value Problems ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Constant Sign ◽  
Dirichlet Boundary Value Problem ◽  
Multiplicity Results ◽  
Sign Changing Solutions ◽  
Exact Multiplicity

Using bifurcation techniques and Sturm comparison theorem, we establish exact multiplicity results of sign-changing or constant sign solutions for the boundary value problemsu″+a(t)f(u)=0,t∈(0,1),u(0)=0, andu(1)=0, wheref∈C(ℝ,ℝ)satisfiesf(0)=0and the limitsf∞=lim|s|→∞(f(s)/s),f0=lim|s|→0(f(s)/s)∈{0,∞}. Weight functiona(t)∈C1[0,1]satisfiesa(t)>0on[0,1].

scholarly journals Existence and uniqueness results for fractional Navier boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Imed Bachar ◽  
Hassan Eltayeb
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Operator Equation ◽  
Real Function ◽  
Existence And Uniqueness ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Existence And Uniqueness Results ◽  
Uniqueness Results ◽  
Alpha Beta

Abstract We establish the existence, uniqueness, and positivity for the fractional Navier boundary value problem: $$\begin{aligned} \textstyle\begin{cases} D^{\alpha }(D^{\beta }\omega )(t)=h(t,\omega (t),D^{\beta }\omega (t)), & 0< t< 1, \\ \omega (0)=\omega (1)=D^{\beta }\omega (0)=D^{\beta }\omega (1)=0, \end{cases}\displaystyle \end{aligned}$$ { D α ( D β ω ) ( t ) = h ( t , ω ( t ) , D β ω ( t ) ) , 0 < t < 1 , ω ( 0 ) = ω ( 1 ) = D β ω ( 0 ) = D β ω ( 1 ) = 0 , where $\alpha,\beta \in (1,2]$ α , β ∈ ( 1 , 2 ] , $D^{\alpha }$ D α and $D^{\beta }$ D β are the Riemann–Liouville fractional derivatives. The nonlinear real function h is supposed to be continuous on $[0,1]\times \mathbb{R\times R}$ [ 0 , 1 ] × R × R and satisfy appropriate conditions. Our approach consists in reducing the problem to an operator equation and then applying known results. We provide an approximation of the solution. Our results extend those obtained in (Dang et al. in Numer. Algorithms 76(2):427–439, 2017) to the fractional setting.

scholarly journals Existence theory for single and multiple solutions to semipositone discrete Dirichlet boundary value problems with singular dependent nonlinearities

2003 ◽  
Vol 16 (1) ◽  
pp. 19-31 ◽  
Author(s):  
Daqing Jiang ◽  
Lili Zhang ◽  
Donal O'Regan ◽  
Ravi P. Agarwal
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Multiple Solutions ◽  
Dirichlet Boundary ◽  
Nonlinear Term ◽  
Boundary Value ◽  
Existence Theory ◽  
Dirichlet Boundary Value Problem

In this paper we establish the existence of single and multiple solutions to the semipositone discrete Dirichlet boundary value problem {Δ2y(i−1)+μf(i,y(i))=0,            i∈{1,2,…,T}y(0)=y(T+1)=0, where μ>0 is a constant and our nonlinear term f(i,u) may be singular at u=0.

scholarly journals Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1907
Author(s):  
Valery Karachik
Keyword(s):  
Boundary Value Problem ◽  
Unit Ball ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Dirichlet Boundary Value Problem ◽  
Polyharmonic Equation ◽  
Harmonic Components

In the previous author’s works, a representation of the solution of the Dirichlet boundary value problem for the biharmonic equation in terms of Green’s function is found, and then it is shown that this representation for a ball can be written in the form of the well-known Almansi formula with explicitly defined harmonic components. In this paper, this idea is extended to the Dirichlet boundary value problem for the polyharmonic equation, but without invoking the Green’s function. It turned out to find an explicit representation of the harmonic components of the m-harmonic function, which is a solution to the Dirichlet boundary value problem, in terms of m solutions to the Dirichlet boundary value problems for the Laplace equation in the unit ball. Then, using this representation, an explicit formula for the harmonic components of the solution to the Neumann boundary value problem for the polyharmonic equation in the unit ball is obtained. Examples are given that illustrate all stages of constructing solutions to the problems under consideration.

On the solvability of the degenerate Noetherian difference-algebraic boundary value problem

2019 ◽  
Vol 33 ◽  
pp. 204-217
Author(s):  
Sergei Chuiko ◽  
Yaroslav Kalinichenko ◽  
Nikita Popov
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Difference Equations ◽  
Nonlinear Oscillations ◽  
Bounded Operator ◽  
Boundary Value ◽  
Homogeneous Part ◽  
Solvability Conditions ◽  
Engineering Control ◽  
Generalized Green Operator

The original conditions of solvability and the scheme of finding solutions of a linear Noetherian difference-algebraic boundary-value problem are proposed in the article, while the technique of pseudoinversion of matrices by Moore-Penrose is substantially used. The problem posed in the article continues to study the conditions for solvability of linear Noetherian boundary value problems given in the monographs of A.M. Samoilenko, A.V. Azbelev, V.P. Maximov, L.F. Rakhmatullina and A.A. Boichuk. The study of differential-algebraic boundary-value problems is closely related to the investigation of boundary-value problems for difference equations, initiated in the works of A.A. Markov, S.N. Bernstein, Y.S. Bezikovych, O.O. Gelfond, S.L. Sobolev, V.S. Ryabenkyi, V.B. Demidovych, A. Halanai, G.I. Marchuk, A.A. Samarskyi, Yu.A. Mytropolskyi, D.I. Martyniuk, G.M. Vainiko, A.M. Samoilenko and A.A. Boichuk. On the other hand, the study of boundary-value problems for difference equations is related to the study of differential-algebraic boundary-value problems initiated in the papers of K. Weierstrass, N.N. Lusin and F.R. Gantmacher. Systematic study of differential-algebraic boundary value problems is devoted to the works of S. Campbell, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, N.A. Perestiyk, V.P. Yakovets, A.A. Boichuk, A. Ilchmann and T. Reis. The study of differential-algebraic boundary value problems is also associated with numerous applications of such problems in the theory of nonlinear oscillations, in mechanics, biology, radio engineering, control theory, motion stability theory. The general case of a linear bounded operator corresponding to the homogeneous part of a linear Noetherian difference-algebraic boundary value problem has no inverse is investigated. The generalized Green operator of a linear difference-algebraic boundary value problem is constructed in the article. The relevance of the study of solvability conditions, as well as finding solutions of linear Noetherian difference-algebraic boundary-value problems, is associated with the widespread use of difference-algebraic boundary-value problems obtained by linearizing nonlinear Noetherian boundary-value problems for systems of ordinary differential and difference equations. Solvability conditions are proposed, as well as the scheme of finding solutions of linear Noetherian difference-algebraic boundary value problems are illustrated in detail in the examples.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

scholarly journals Existence of Multiple Positive Solutions for Even Order Multi-Point Boundary Value Problems

2007 ◽  
Vol 14 (4) ◽  
pp. 775-792
Author(s):  
Youyu Wang ◽  
Weigao Ge
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Positive Solutions ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Multiple Positive Solutions ◽  
Point Boundary ◽  
Even Order

Abstract In this paper, we consider the existence of multiple positive solutions for the 2𝑛th order 𝑚-point boundary value problem: where (0,1), 0 < ξ 1 < ξ 2 < ⋯ < ξ 𝑚–2 < 1. Using the Leggett–Williams fixed point theorem, we provide sufficient conditions for the existence of at least three positive solutions to the above boundary value problem. The associated Green's function for the above problem is also given.

scholarly journals A study on multiterm hybrid multi-order fractional boundary value problem coupled with its stability analysis of Ulam–Hyers type

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Ahmed Nouara ◽  
Abdelkader Amara ◽  
Eva Kaslik ◽  
Sina Etemad ◽  
Shahram Rezapour ◽  
...  
Keyword(s):  
Stability Analysis ◽  
Boundary Value Problem ◽  
Fractional Derivatives ◽  
Boundary Value ◽  
Research Work ◽  
Existence Results ◽  
Fractional Boundary Value Problem ◽  
Numerical Examples ◽  
Fractional Derivatives And Integrals ◽  
Fractional Differential

AbstractIn this research work, a newly-proposed multiterm hybrid multi-order fractional boundary value problem is studied. The existence results for the supposed hybrid fractional differential equation that involves Riemann–Liouville fractional derivatives and integrals of multi-orders type are derived using Dhage’s technique, which deals with a composition of three operators. After that, its stability analysis of Ulam–Hyers type and the relevant generalizations are checked. Some illustrative numerical examples are provided at the end to illustrate and validate our obtained results.

scholarly journals Existence results for first derivative dependent ϕ-Laplacian boundary value problems

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Imran Talib ◽  
Thabet Abdeljawad
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Continuous Functions ◽  
Existence Of Solution ◽  
Existence Results ◽  
Main Concern ◽  
First Derivative ◽  
Boundary Functions ◽  
Carathéodory Function

Abstract Our main concern in this article is to investigate the existence of solution for the boundary-value problem $$\begin{aligned}& (\phi \bigl(x'(t)\bigr)'=g_{1} \bigl(t,x(t),x'(t)\bigr),\quad \forall t\in [0,1], \\& \Upsilon _{1}\bigl(x(0),x(1),x'(0)\bigr)=0, \\& \Upsilon _{2}\bigl(x(0),x(1),x'(1)\bigr)=0, \end{aligned}$$ ( ϕ ( x ′ ( t ) ) ′ = g 1 ( t , x ( t ) , x ′ ( t ) ) , ∀ t ∈ [ 0 , 1 ] , ϒ 1 ( x ( 0 ) , x ( 1 ) , x ′ ( 0 ) ) = 0 , ϒ 2 ( x ( 0 ) , x ( 1 ) , x ′ ( 1 ) ) = 0 , where $g_{1}:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}$ g 1 : [ 0 , 1 ] × R 2 → R is an $L^{1}$ L 1 -Carathéodory function, $\Upsilon _{i}:\mathbb{R}^{3}\rightarrow \mathbb{R} $ ϒ i : R 3 → R are continuous functions, $i=1,2$ i = 1 , 2 , and $\phi :(-a,a)\rightarrow \mathbb{R}$ ϕ : ( − a , a ) → R is an increasing homeomorphism such that $\phi (0)=0$ ϕ ( 0 ) = 0 , for $0< a< \infty $ 0 < a < ∞ . We obtain the solvability results by imposing some new conditions on the boundary functions. The new conditions allow us to ensure the existence of at least one solution in the sector defined by well ordered functions. These ordered functions do not require one to check the definitions of lower and upper solutions. Moreover, the monotonicity assumptions on the arguments of boundary functions are not required in our case. An application is considered to ensure the applicability of our results.

Sign in / Sign up

Export Citation Format

Share Document