scholarly journals Existence and Iterative Method for Some Riemann Fractional Nonlinear Boundary Value Problems

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 961 ◽  
Author(s):  
Imed Bachar ◽  
Habib Mâagli ◽  
Hassan Eltayeb
Keyword(s):  
Iterative Method ◽  
Boundary Value Problems ◽  
Fourth Order ◽  
Existence And Uniqueness ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Uniqueness Of Solution ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
Appl Math

In this paper, we prove the existence and uniqueness of solution for some Riemann–Liouville fractional nonlinear boundary value problems. The positivity of the solution and the monotony of iterations are also considered. Some examples are presented to illustrate the main results. Our results generalize those obtained by Wei et al (Existence and iterative method for some fourth order nonlinear boundary value problems. Appl. Math. Lett. 2019, 87, 101–107.) to the fractional setting.

scholarly journals Existence and uniqueness of solutions for a class of fractional nonlinear boundary value problems under mild assumptions

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Imed Bachar ◽  
Habib Mâagli ◽  
Hassan Eltayeb
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Existence And Uniqueness ◽  
Nonlinear Boundary ◽  
Continuous Solution ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problems ◽  
Uniqueness Of Solutions ◽  
Nonlinear Boundary Value ◽  
Appl Math

AbstractWe deal with the following Riemann–Liouville fractional nonlinear boundary value problem: $$ \textstyle\begin{cases} \mathcal{D}^{\alpha }v(x)+f(x,v(x))=0, & 2< \alpha \leq 3, x\in (0,1), \\ v(0)=v^{\prime }(0)=v(1)=0. \end{cases} $$ { D α v ( x ) + f ( x , v ( x ) ) = 0 , 2 < α ≤ 3 , x ∈ ( 0 , 1 ) , v ( 0 ) = v ′ ( 0 ) = v ( 1 ) = 0 . Under mild assumptions, we prove the existence of a unique continuous solution v to this problem satisfying $$ \bigl\vert v(x) \bigr\vert \leq cx^{\alpha -1}(1-x)\quad\text{for all }x \in [ 0,1]\text{ and some }c>0. $$ | v ( x ) | ≤ c x α − 1 ( 1 − x ) for all  x ∈ [ 0 , 1 ]  and some  c > 0 . Our results improve those obtained by Zou and He (Appl. Math. Lett. 74:68–73, 2017).

scholarly journals Solving Nonlinear Fourth-Order Boundary Value Problems Using a Numerical Approach: (m+1)th-Step Block Method

2017 ◽  
Vol 2017 ◽  
pp. 1-9 ◽  
Author(s):  
Oluwaseun Adeyeye ◽  
Zurni Omar
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Boundary Value ◽  
Numerical Approach ◽  
Squeezing Flow ◽  
Nonlinear Boundary Value Problems ◽  
Block Method ◽  
Nonlinear Boundary Value ◽  
Improved Accuracy

Nonlinear boundary value problems (BVPs) are more tedious to solve than their linear counterparts. This is observed in the extra computation required when determining the missing conditions in transforming BVPs to initial value problems. Although a number of numerical approaches are already existent in literature to solve nonlinear BVPs, this article presents a new block method with improved accuracy to solve nonlinear BVPs. A m+1th-step block method is developed using a modified Taylor series approach to directly solve fourth-order nonlinear boundary value problems (BVPs) where m is the order of the differential equation under consideration. The schemes obtained were combined to simultaneously produce solution to the fourth-order nonlinear BVPs at m+1 points iteratively. The derived block method showed improved accuracy in comparison to previously existing authors when solving the same problems. In addition, the suitability of the m+1th-step block method was displayed in the solution for magnetohydrodynamic squeezing flow in porous medium.

scholarly journals Nonresonance conditions for fourth order nonlinear boundary value problems

1994 ◽  
Vol 17 (4) ◽  
pp. 725-740 ◽  
Author(s):  
C. De Coster ◽  
C. Fabry ◽  
F. Munyamarere
Keyword(s):  
Boundary Value Problems ◽  
Fourth Order ◽  
Nonlinear Boundary ◽  
Linear Problem ◽  
Bounded Function ◽  
Boundary Value ◽  
Nonlinear Boundary Value Problems ◽  
Nonlinear Boundary Value ◽  
Existence Conditions

This paper is devoted to the study of the problemu(4)=f(t,u,u′,u″,u‴),u(0)=u(2π),   u′(0)=u′(2π),   u″(0)=u″(2π),   u‴(0)=u‴(2π).We assume thatfcan be written under the formf(t,u,u′,u″,u‴)=f2(t,u,u′,u″,u‴)u″+f1(t,u,u′,u″,u‴)u′+f0(t,u,u′,u″,u‴)u+r(t,u,u′,u″,u‴)whereris a bounded function. We obtain existence conditions related to uniqueness conditions for the solution of the linear problemu(4)=au+bu″,u(0)=u(2π),   u′(0)=u′(2π),   u″(0)=u″(2π),   u‴(0)=u‴(2π).

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