Abstract Algebraic Construction in Fractional Calculus: Parametrised Families with Semigroup Properties

Fernandez, Arran

doi:10.1007/s11785-024-01493-6

Complex Anal. Oper. Theory

2024

DOI: 10.1007/s11785-024-01493-6

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Abstract Algebraic Construction in Fractional Calculus: Parametrised Families with Semigroup Properties

Arran Fernandez

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2024

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Cited by 4 publications

References 28 publications

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Searching for Sonin kernels

Ortigueira

2024

Fract Calc Appl Anal

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The causal shift-invariant convolution is studied from the point of view of inversion. Abel’s algorithm, used in the tautochrone problem, is considered and Sonin’s existence condition is deduced. To generate pairs of functions verifying Sonin’s condition, the class of Mittag-Leffler type functions is used. In particular, functions that are impulse responses of ARMA(N,N) systems serve as a basis. The possible use of Abel’s procedure as a support for introducing generalized fractional derivatives is evaluated.

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Searching for Sonin kernels

Ortigueira

2024

Fract Calc Appl Anal

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Non-Additivity and Additivity in General Fractional Calculus and Its Physical Interpretations

Tarasov

2024

Fractal Fract

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In this work, some properties of the general convolutional operators of general fractional calculus (GFC), which satisfy analogues of the fundamental theorems of calculus, are described. Two types of general fractional (GF) operators on a finite interval exist in GFC that are conventionally called the L-type and T-type operators. The main difference between these operators is that the additivity property holds for T-type operators and is violated for L-type operators. This property is very important for the application of GFC in physics and other sciences. The presence or violation of the additivity property can be associated with qualitative differences in the behavior of physical processes and systems. In this paper, we define L-type line GF integrals and L-type line GF gradients. For these L-type operators, the gradient theorem is proved in this paper. In general, the L-type line GF integral over a simple line is not equal to the sum of the L-type line GF integrals over lines that make up the entire line. In this work, it is shown that there exist two cases when the additivity property holds for the L-type line GF integrals. In the first case, the L-type line GF integral along the line is equal to the sum of the L-type line GF integrals along parts of this line only if the processes, which are described by these lines, are independent. Processes are called independent if the history of changes in the subsequent process does not depend on the history of the previous process. In the second case, we prove the additivity property holds for the L-type line GF integrals, if the conditions of the GF gradient theorems are satisfied.

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General Fractional Economic Dynamics with Memory

Tarasov

2024

Mathematics

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For the first time, a self-consistent mathematical approach to describe economic processes with a general form of a memory function is proposed. In this approach, power-type memory is a special case of such general memory. The memory is described by pairs of memory functions that satisfy the Sonin and Luchko conditions. We propose using general fractional calculus (GFC) as a mathematical language that allows us to describe a general form of memory in economic processes. The existence of memory (non-locality in time) means that the process depends on the history of changes to this process in the past. Using GFC, exactly solvable economic models of natural growth with a general form of memory are proposed. Equations of natural growth with general memory are equations with general fractional derivatives and general fractional integrals for which the fundamental theorems of GFC are satisfied. Exact solutions for these equations of models of natural growth with general memory are derived. The properties of dynamic maps with a general form of memory are described in the general form and do not depend on the choice of specific types of memory functions. Examples of these solutions for various types of memory functions are suggested.

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Abstract Algebraic Construction in Fractional Calculus: Parametrised Families with Semigroup Properties

Cited by 4 publications

References 28 publications

Searching for Sonin kernels

Searching for Sonin kernels

Non-Additivity and Additivity in General Fractional Calculus and Its Physical Interpretations

General Fractional Economic Dynamics with Memory

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