Approximate Solutions of Equations in Abstract g-Fractional Calculus

2017 ◽  
pp. 121-137
Author(s):  
George A. Anastassiou ◽  
Ioannis K. Argyros
Keyword(s):  
Fractional Calculus ◽  
Approximate Solutions ◽  
Solutions Of Equations

scholarly journals General Fractional Dynamics

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1464
Author(s):  
Vasily E. Tarasov
Keyword(s):  
Fractional Calculus ◽  
Exact Solutions ◽  
Arbitrary Order ◽  
Nonlinear Dynamical Systems ◽  
Fractional Integrals ◽  
Fractional Dynamics ◽  
Fractional Systems ◽  
Nonlinear Dynamical ◽  
Fractional Differential ◽  
Solutions Of Equations

General fractional dynamics (GFDynamics) can be viewed as an interdisciplinary science, in which the nonlocal properties of linear and nonlinear dynamical systems are studied by using general fractional calculus, equations with general fractional integrals (GFI) and derivatives (GFD), or general nonlocal mappings with discrete time. GFDynamics implies research and obtaining results concerning the general form of nonlocality, which can be described by general-form operator kernels and not by its particular implementations and representations. In this paper, the concept of “general nonlocal mappings” is proposed; these are the exact solutions of equations with GFI and GFD at discrete points. In these mappings, the nonlocality is determined by the operator kernels that belong to the Sonin and Luchko sets of kernel pairs. These types of kernels are used in general fractional integrals and derivatives for the initial equations. Using general fractional calculus, we considered fractional systems with general nonlocality in time, which are described by equations with general fractional operators and periodic kicks. Equations with GFI and GFD of arbitrary order were also used to derive general nonlocal mappings. The exact solutions for these general fractional differential and integral equations with kicks were obtained. These exact solutions with discrete timepoints were used to derive general nonlocal mappings without approximations. Some examples of nonlocality in time are described.

scholarly journals Approximate solutions of equations defined by complex spherical multiplier operators

2005 ◽  
Vol 2005 (2) ◽  
pp. 93-115
Author(s):  
C. P. Oliveira
Keyword(s):  
Approximate Solutions ◽  
Positive Definite ◽  
Positive Definite Functions ◽  
Sobolev Type ◽  
Sobolev Type Spaces ◽  
Type Spaces ◽  
Solutions Of Equations ◽  
Multiplier Operators

This paper studies, in a partial but concise manner, approximate solutions of equations defined by complex spherical multiplier operators. The approximations are from native spaces embedded in Sobolev-type spaces and derived from the use of positive definite functions to perform spherical interpolation.

Semi-local convergence of a Newton-like method for solving equations with a singular derivative

2018 ◽  
Vol 27 (1) ◽  
pp. 01-08
Author(s):  
IOANNIS K. ARGYROS ◽  
◽  
GEORGE SANTHOSH ◽  
Keyword(s):  
Banach Space ◽  
Convergence Analysis ◽  
Real Line ◽  
Local Convergence ◽  
Approximate Solutions ◽  
Convergence Domain ◽  
Solving Equations ◽  
Solutions Of Equations ◽  
Local Convergence Analysis ◽  
Special Case

We present a semi-local convergence analysis for a Newton-like method to approximate solutions of equations when the derivative is not necessarily non-singular in a Banach space setting. In the special case when the equation is defined on the real line the convergence domain is improved for this method when compared to earlier results. Numerical results where earlier results cannot apply but the new results can apply to solve nonlinear equations are also presented in this study.

scholarly journals Approximate solutions of one-dimensional systems with fractional derivative

2020 ◽  
Vol 31 (07) ◽  
pp. 2050092
Author(s):  
A. Ferrari ◽  
M. Gadella ◽  
L. P. Lara ◽  
E. Santillan Marcus
Keyword(s):  
Fractional Calculus ◽  
Fractional Derivative ◽  
Fractional Differential Equations ◽  
Caputo Derivative ◽  
Approximate Solutions ◽  
One Dimensional ◽  
Dissipative Term ◽  
Numerical Resolution ◽  
Fractional Caputo Derivative ◽  
Fractional Differential

The fractional calculus is useful to model nonlocal phenomena. We construct a method to evaluate the fractional Caputo derivative by means of a simple explicit quadratic segmentary interpolation. This method yields to numerical resolution of ordinary fractional differential equations. Due to the nonlocality of the fractional derivative, we may establish an equivalence between fractional oscillators and ordinary oscillators with a dissipative term.

scholarly journals Global strong solutions of equations of magnetohydrodynamic type

1997 ◽  
Vol 38 (3) ◽  
pp. 291-306 ◽  
Author(s):  
Marko A. Rojas-Medar ◽  
José Luiz Boldrini
Keyword(s):  
Global Existence ◽  
Galerkin Method ◽  
Error Bounds ◽  
Strong Solutions ◽  
Approximate Solutions ◽  
System Of Equations ◽  
Spectral Galerkin Method ◽  
Global Strong Solutions ◽  
Solutions Of Equations

AbstractBy using the spectral Galerkin method, we prove a result on the global existence in time of strong solutions for a system of equations of magnetohydrodynamic type. Several estimates for the solution and their approximations are given. These estimates can be used in the derivation of error bounds for the approximate solutions.

Spectral-Collocation Method for Volterra Delay Integro-Differential Equations with Weakly Singular Kernels

2016 ◽  
Vol 8 (4) ◽  
pp. 648-669 ◽  
Author(s):  
Xiulian Shi ◽  
Yanping Chen
Keyword(s):  
Differential Equations ◽  
Error Analysis ◽  
Approximate Solutions ◽  
Weakly Singular Kernels ◽  
Numerical Examples ◽  
Weakly Singular ◽  
Singular Kernels ◽  
Solutions Of Equations ◽  
Rigorous Error ◽  
Special Case

AbstractA spectral Jacobi-collocation approximation is proposed for Volterra delay integro-differential equations with weakly singular kernels. In this paper, we consider the special case that the underlying solutions of equations are sufficiently smooth. We provide a rigorous error analysis for the proposed method, which shows that both the errors of approximate solutions and the errors of approximate derivatives decay exponentially inL∞norm and weightedL2norm. Finally, two numerical examples are presented to demonstrate our error analysis.

scholarly journals Magnetic field diffusion in ferromagnetic materials: fractional calculus approaches

2021 ◽  
Vol 11 (3) ◽  
pp. 1-15
Author(s):  
Jordan Hristov
Keyword(s):  
Magnetic Field ◽  
Fractional Calculus ◽  
Approximate Solutions ◽  
Ferromagnetic Materials ◽  
Fading Memory ◽  
Dirichlet Problems ◽  
Integration Model ◽  
Parabolic Profile ◽  
Field Diffusion ◽  
Singular Memory

The paper addresses diffusion approximations of magnetic field penetration of ferromagnetic materials with emphasis on fractional calculus applications and relevant approximate solutions. Examples with applications of time-fractional semi-derivatives and singular kernel models (Caputo time fractional operator) in cases of field independent and field-dependent magnetic diffusivities have been developed: Dirichlet problems and time-dependent boundary condition (power-law ramp). Approximate solutions in all theses case have been developed by applications of the integral-balance method and assumed parabolic profile with unspecified exponents. Tow version of the integral method have been successfully implemented: SDIM (single integration applicable to time-fractional semi-derivative model) and DIM (double-integration model to fractionalized singular memory models). The fading memory approach in the sense of the causality concept and memory kernel effect on the model constructions have been discussed.

scholarly journals An Efficient Mechanism to Solve Fractional Differential Equations Using Fractional Decomposition Method

Symmetry ◽  
2021 ◽  
Vol 13 (6) ◽  
pp. 984
Author(s):  
Mahmoud S. Alrawashdeh ◽  
Seba A. Migdady ◽  
Ioannis K. Argyros
Keyword(s):  
Differential Equations ◽  
Fractional Calculus ◽  
Laplace Transform ◽  
Fractional Differential Equations ◽  
Decomposition Method ◽  
Large Scale ◽  
Approximate Solutions ◽  
Science And Engineering ◽  
Large Scale Problems ◽  
Fractional Differential

We present some new results that deal with the fractional decomposition method (FDM). This method is suitable to handle fractional calculus applications. We also explore exact and approximate solutions to fractional differential equations. The Caputo derivative is used because it allows traditional initial and boundary conditions to be included in the formulation of the problem. This is of great significance for large-scale problems. The study outlines the significant features of the FDM. The relation between the natural transform and Laplace transform is a symmetrical one. Our work can be considered as an alternative to existing techniques, and will have wide applications in science and engineering fields.

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