Many different types of fractional calculus have been defined, which may be categorised into broad classes according to their properties and behaviours. Two types that have been much studied in the ...literature are the Hadamard-type fractional calculus and tempered fractional calculus. This paper establishes a connection between these two definitions, writing one in terms of the other by making use of the theory of fractional calculus with respect to functions. By extending this connection in a natural way, a generalisation is developed which unifies several existing fractional operators: Riemann–Liouville, Caputo, classical Hadamard, Hadamard-type, tempered, and all these taken with respect to functions. The fundamental calculus of these generalised operators is established, including semigroup and reciprocal properties as well as application to some example functions. Function spaces are constructed in which the new operators are defined and bounded. Finally, some formulae are derived for fractional integration by parts with these operators.
Physical parameters in dimensionless form in the governing equations of real-life phenomena naturally occur. How to control them by determining their range of validity is in general a big issue. In ...this paper, a mathematical approach is presented to identify the correct range of physical parameters adopting the recently popular analytic approximate Adomian decomposition method (ADM). Having found the approximate analytical Adomian series solution up to a specified truncation order, the squared residual error formula is employed to work out the threshold and the existence domain of certain physical parameters satisfying a preassigned tolerance. If the current procedure is not closely pursued, the presented results with the ADM may not be up to the desired level of accuracy (the worst is the divergent physically meaningless solutions), or much more ADM series terms need to be computed to satisfy certain accuracy. Examples reveal the necessity of the present approach to make sure that the results embark the correct range of physical parameters in the study of a physical problem containing several dominating parameters.
In this paper, we study two classes of Kirchhoff-type problems set on a double-phase framework. That is, the functional space where finding solutions coincides with the Musielak–Orlicz–Sobolev space
...W
0
1
,
H
(
Ω
)
, with modular function
H
related to the so-called double-phase operator. Via a variational approach, we provide existence and multiplicity results.
The object of the present paper is to characterize the class of Kenmotsu manifolds which admits conformal
η
-Ricci soliton. Here, we have investigated the nature of the conformal
η
-Ricci soliton ...within the framework of Kenmotsu manifolds. It is shown that an
η
-Einstein Kenmotsu manifold admitting conformal
η
-Ricci soliton is an Einstein one. Moving further, we have considered gradient conformal
η
-Ricci soliton on Kenmotsu manifold and established a relation between the potential vector field and the Reeb vector field. Next, it is proved that under certain condition, a conformal
η
-Ricci soliton on Kenmotsu manifolds under generalized D-conformal deformation remains invariant. Finally, we have constructed an example for the existence of conformal
η
-Ricci soliton on Kenmotsu manifold.
In this paper, we study finite difference methods for fractional differential equations (FDEs) with Caputo–Hadamard derivatives. First, smoothness properties of the solution are investigated. The ...fractional rectangular,
L
log
,
1
interpolation, and modified predictor–corrector methods for Caputo–Hadamard fractional ordinary differential equations (FODEs) are proposed through approximating the corresponding equivalent Volterra integral equations. The stability and error estimate of the derived methods are proved as well. Then, we investigate finite difference methods for fractional partial differential equations (FPDEs) with Caputo–Hadamard derivative. By applying the constructed
L
1 scheme for approximating the time fractional derivative, a semi-discrete difference scheme is derived. The stability and convergence analysis are shown too in detail. Furthermore, a fully discrete scheme is established by the standard second-order difference scheme in spacial direction. Stability and error estimate are also presented. The numerical experiments are displayed to verify the theoretical results.
In this work, we give the general framework for constructing trigonometrically fitted symplectic Runge–Kutta–Nyström (RKN) methods from symplectic trigonometrically fitted partitioned Runge–Kutta ...(PRK) methods. We construct RKN methods from PRK methods with up to five stages and fourth algebraic order. Numerical results are given for the two-body problem and the perturbed two-body problem.
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