Iterative techniques with computer realization for the initial value problem for Caputo fractional differential equations

2017 ◽  
Vol 58 (1-2) ◽  
pp. 433-467 ◽  
Author(s):  
R. Agarwal ◽  
A. Golev ◽  
S. Hristova ◽  
D. O’Regan ◽  
K. Stefanova
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Initial Value ◽  
Iterative Techniques ◽  
Computer Realization ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

A superlinear convergence scheme for nonlinear fractional differential equations and its fast implement

2019 ◽  
Vol 10 (01) ◽  
pp. 1941001 ◽  
Author(s):  
Haobo Gong ◽  
Jingna Zhang ◽  
Hao Guo ◽  
Jianfei Huang
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Superlinear Convergence ◽  
Initial Value ◽  
Computational Performance ◽  
Caputo Fractional Differential Equations ◽  
Long Time ◽  
Efficient Scheme ◽  
Fractional Differential ◽  
Theoretical Results

In this paper, we first construct an efficient scheme for nonlinear Caputo fractional differential equations with the initial value and the fractional degree [Formula: see text]. Then, the unconditional stability and the superlinear convergence with the order [Formula: see text] of the proposed scheme are strictly proved and discussed. Due to the nonlocal property of fractional operators, the new scheme is time-consuming for long-time simulations. Thus, a fast implement of the proposed scheme is presented based on the sum-of-exponentials (SOE) approximation for the kernel [Formula: see text] on the interval [Formula: see text] in the Riemann–Liouville integral, where [Formula: see text] is the stepsize. Some numerical experiments are provided to support the theoretical results of the new scheme and demonstrate the computational performance of its fast implement.

scholarly journals Initial Value Problem For Nonlinear Fractional Differential Equations With ψ-Caputo Derivative Via Monotone Iterative Technique

Axioms ◽  
2020 ◽  
Vol 9 (2) ◽  
pp. 57 ◽  
Author(s):  
Choukri Derbazi ◽  
Zidane Baitiche ◽  
Mouffak Benchohra ◽  
Alberto Cabada
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Caputo Derivative ◽  
Monotone Iterative Technique ◽  
Extremal Solutions ◽  
Iterative Technique ◽  
Initial Value ◽  
Monotone Iterative ◽  
Fractional Differential

In this article, we discuss the existence and uniqueness of extremal solutions for nonlinear initial value problems of fractional differential equations involving the ψ -Caputo derivative. Moreover, some uniqueness results are obtained. Our results rely on the standard tools of functional analysis. More precisely we apply the monotone iterative technique combined with the method of upper and lower solutions to establish sufficient conditions for existence as well as the uniqueness of extremal solutions to the initial value problem. An illustrative example is presented to point out the applicability of our main results.

scholarly journals Research On different kinds of arcs in interval valued fuzzy graphs Ann Mary Philip 1 , Sunny Joseph Kalayathankal 2 and Joseph Varghese Kureethara 3* Malaya Journal of Matematik : Volume 7, Issue 2, 2019, Pages:309-313 DOI: 10.26637/MJM0702/0025 | Full Article PDF | PDF size (1,392.80 KB) Research Initial-value problems for nonlinear hybrid implicit Caputo fractional differential equations Abdelouaheb Ardjouni 1 * and Ahcene Djoudi 2 Malaya Journal of Matematik : Volume 7, Issue 2, 2019, Pages:314-317 DOI: 10.26637/MJM0702/0026 | Full Article PDF | PDF size (1,390.08 KB) Research Some new results on the connected sum of certain digital surfaces Ismet Cinar 1 * and Ismet KARACA 2 Malaya Journal of Matematik : Volume 7, Issue 2, 2019, Pages:318-325 DOI: 10.26637/MJM0702/0027 | Full Article PDF | PDF size (1,562.13 KB) Research A frictionless contact problem for elastic-viscoplastic materials with adhesion and thermal effects Tedjani Hadj Ammar 1 * and Khezzani Rimi 2 Malaya Journal of Matematik : Volume 7, Issue 2, 2019, Pages:326-337 DOI: 10.26637/MJM0702/0028 | Full Article PDF | PDF size (1,510.23 KB) Research Convergence analysis and approximate solution of fractional differential equations

2019 ◽  
Vol 7 (2) ◽  
pp. 338-344
Author(s):  
Bhausaheb R. Sontakke ◽  
Rajashri Pandit
Keyword(s):  
Full Article ◽  
Differential Equations ◽  
Fractional Differential Equations ◽  
Thermal Effects ◽  
Initial Value ◽  
Fuzzy Graphs ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
Interval Valued ◽  
Nonlinear Hybrid

On coupled system of nonlinear Ψ-Hilfer hybrid fractional differential equations

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ashwini D. Mali ◽  
Kishor D. Kucche ◽  
José Vanterler da Costa Sousa
Keyword(s):  
Fixed Point ◽  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Initial Value Problem ◽  
Fixed Point Theorems ◽  
Existence Of Solutions ◽  
Coupled System ◽  
Initial Value ◽  
Fractional Differential

Abstract This paper is dedicated to investigating the existence of solutions to the initial value problem (IVP) for a coupled system of Ψ-Hilfer hybrid fractional differential equations (FDEs) and boundary value problem (BVP) for a coupled system of Ψ-Hilfer hybrid FDEs. Analysis of the current paper depends on the two fixed point theorems involving three operators characterized on Banach algebra. In the view of an application, we provided useful examples to exhibit the effectiveness of our achieved results.

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