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 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

Caputo fractional differential equations with non-instantaneous impulses and strict stability by Lyapunov functions

Filomat ◽  
2017 ◽  
Vol 31 (16) ◽  
pp. 5217-5239 ◽  
Author(s):  
Ravi Agarwal ◽  
Snehana Hristova ◽  
Donal O’Regan
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Time Intervals ◽  
Dini Derivative ◽  
Comparison Results ◽  
Strict Stability ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential

In this paper the statement of initial value problems for fractional differential equations with noninstantaneous impulses is given. These equations are adequate models for phenomena that are characterized by impulsive actions starting at arbitrary fixed points and remaining active on finite time intervals. Strict stability properties of fractional differential equations with non-instantaneous impulses by the Lyapunov approach is studied. An appropriate definition (based on the Caputo fractional Dini derivative of a function) for the derivative of Lyapunov functions among the Caputo fractional differential equations with non-instantaneous impulses is presented. Comparison results using this definition and scalar fractional differential equations with non-instantaneous impulses are presented and sufficient conditions for strict stability and uniform strict stability are given. Examples are given to illustrate the theory.

scholarly journals Implicit nonlinear fractional differential equations of variable order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Amar Benkerrouche ◽  
Mohammed Said Souid ◽  
Kanokwan Sitthithakerngkiet ◽  
Ali Hakem
Keyword(s):  
Differential Equations ◽  
Boundary Value Problem ◽  
Fractional Differential Equations ◽  
Boundary Value ◽  
Variable Order ◽  
Stability Of Solutions ◽  
Nonlinear Fractional Differential Equations ◽  
Caputo Fractional Differential Equations ◽  
Fractional Differential ◽  
The Stability

AbstractIn this manuscript, we examine both the existence and the stability of solutions to the implicit boundary value problem of Caputo fractional differential equations of variable order. We construct an example to illustrate the validity of the observed results.

scholarly journals New Gronwall-Bellman Type Inequalities and Applications in the Analysis for Solutions to Fractional Differential Equations

2013 ◽  
Vol 2013 ◽  
pp. 1-13 ◽  
Author(s):  
Bin Zheng ◽  
Qinghua Feng
Keyword(s):  
Quantitative Analysis ◽  
Differential Equations ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Continuous Dependence ◽  
Initial Value ◽  
Liouville Fractional Derivative ◽  
Explicit Bounds ◽  
Fractional Differential ◽  
Differential And Integral Equations

Some new Gronwall-Bellman type inequalities are presented in this paper. Based on these inequalities, new explicit bounds for the related unknown functions are derived. The inequalities established can also be used as a handy tool in the research of qualitative as well as quantitative analysis for solutions to some fractional differential equations defined in the sense of the modified Riemann-Liouville fractional derivative. For illustrating the validity of the results established, we present some applications for them, in which the boundedness, uniqueness, and continuous dependence on the initial value for the solutions to some certain fractional differential and integral equations are investigated.

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