Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid
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Pushpendra Kumar
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Vedat Suat Erturk
Abstract
In this article, we propose generalized forms of three well-known fractional numerical methods namely Euler, Runge–Kutta 2-step, and Runge–Kutta 4-step, respectively. The new versions we provide of these methods are derived by utilizing a non-uniform grid which is slightly different from previous versions of these algorithms. A new generalized form of the well-known Caputo-type fractional derivative is used to derive the results. All necessary analyses related to the stability, convergence, and error bounds are also provided. The precision of all simulated results is justified by performing multiple numerical experiments, with some meaningful problems solved by implementing the code in Mathematica. Finally, we give a brief discussion on the simulated results which shows that the generalized methods are novel, effective, reliable, and very easy to implement.
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Author contribution: Pushpendra Kumar: Investigation, Conceptualization, Methodology, Formal analysis, Visualization, Supervision, Resources, and Writing - original draft. Vedat Suat Erturk: Conceptualization, Methodology, Investigation, Software, Visualization, Supervision, Writing-review, and editing. Marina Murillo-Arcila: Conceptualization, Formal analysis, Visualization, Writing-review, and editing. Charis Harley: Conceptualization, Formal analysis, Visualization, Writing-review, and editing.
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Research funding: The third author is supported by MCIN/AEI/10.13039/501100011033, Project PID2019-105011GBI00, and by Generalitat Valenciana, Project PROMETEU/2021/070.
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Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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Availability of data and materials: All the data is included in the paper.
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© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Frequency responses for induced neural transmembrane potential by electromagnetic waves (1 kHz to 1 GHz)
- Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space
- Optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferential
- Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid
- Controllability of coupled fractional integrodifferential equations
- A new generalized approach to study the existence of solutions of nonlinear fractional boundary value problems
- Rational soliton solutions in the nonlocal coupled complex modified Korteweg–de Vries equations
- Buoyancy driven flow characteristics inside a cavity equiped with diamond elliptic array
- Analysis and numerical effects of time-delayed rabies epidemic model with diffusion
- Two occurrences of fractional actions in nonlinear dynamics
- Multiwave interaction solutions for a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics
- Effects of mixed time delays and D operators on fixed-time synchronization of discontinuous neutral-type neural networks
- Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order
- Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator
- Simulation and modeling of different cell shapes for closed-cell LM-13 alloy foam for compressive behavior
- Pandemic management by a spatio–temporal mathematical model
- Adaptive ADI difference solution of quenching problems based on the 3D convection–reaction–diffusion equation
- Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump
- Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire
- Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects
- Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives
- Improving the dynamic behavior of the plate under supersonic air flow by using nonlinear energy sink
Articles in the same Issue
- Frontmatter
- Original Research Articles
- Frequency responses for induced neural transmembrane potential by electromagnetic waves (1 kHz to 1 GHz)
- Investigating existence results for fractional evolution inclusions with order r ∈ (1, 2) in Banach space
- Optimal control of non-instantaneous impulsive second-order stochastic McKean–Vlasov evolution system with Clarke subdifferential
- Generalized forms of fractional Euler and Runge–Kutta methods using non-uniform grid
- Controllability of coupled fractional integrodifferential equations
- A new generalized approach to study the existence of solutions of nonlinear fractional boundary value problems
- Rational soliton solutions in the nonlocal coupled complex modified Korteweg–de Vries equations
- Buoyancy driven flow characteristics inside a cavity equiped with diamond elliptic array
- Analysis and numerical effects of time-delayed rabies epidemic model with diffusion
- Two occurrences of fractional actions in nonlinear dynamics
- Multiwave interaction solutions for a (3 + 1)-dimensional B-type Kadomtsev–Petviashvili equation in fluid dynamics
- Effects of mixed time delays and D operators on fixed-time synchronization of discontinuous neutral-type neural networks
- Solvability and stability of nonlinear hybrid ∆-difference equations of fractional-order
- Weak and strong boundedness for p-adic fractional Hausdorff operator and its commutator
- Simulation and modeling of different cell shapes for closed-cell LM-13 alloy foam for compressive behavior
- Pandemic management by a spatio–temporal mathematical model
- Adaptive ADI difference solution of quenching problems based on the 3D convection–reaction–diffusion equation
- Null controllability of Hilfer fractional stochastic integrodifferential equations with noninstantaneous impulsive and Poisson jump
- Application of modified Mickens iteration procedure to a pendulum and the motion of a mass attached to a stretched elastic wire
- Modeling the spatiotemporal intracellular calcium dynamics in nerve cell with strong memory effects
- Switched coupled system of nonlinear impulsive Langevin equations involving Hilfer fractional-order derivatives
- Improving the dynamic behavior of the plate under supersonic air flow by using nonlinear energy sink
