This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between ā¦
This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: p+Ī±2-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators.
In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we ā¦
In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we abstracted a family of fractal operators and investigate the kernel function and properties thereof. Based on fractal operators, the intrinsic relation between Bessel function and Struve function was revealed, and some new special functions were found. The results provide mathematical tools for biomechanics and automatic control.
In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be ā¦
In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be characterized by fractal operators, and each type of operator possesses rich invariant properties. Based on the invariant properties of fractal operators, we discovered that the symmetry evolution laws of functional fractal trees in the physical fractal space can reveal the intrinsic correlations between special functions. This article explores the fractional-order correlation between special functions inspired by bone fractal operators. Specifically, the following contents are included: (1) showing the intrinsic expression in the convolutional kernel function of bone fractal operators and its correlation with special functions; (2) proving the following proposition: the convolutional kernel function of bone fractal operators is still related to the special functions under different input signals (external load, external stimulus); (3) using the bone fractal operators as the background and error function as the core, deriving the fractional-order correlation between different special functions.
This paper is dedicated to elucidating the abstract algebraic structure of operational calculus theory. Based on abstract algebra and operational calculus, the operator algebra theory of MikusiÅski has been revised. ā¦
This paper is dedicated to elucidating the abstract algebraic structure of operational calculus theory. Based on abstract algebra and operational calculus, the operator algebra theory of MikusiÅski has been revised. We restate the concept of MikusiÅskiās operator field as a convolutional field and define a new concept of operator field with the differential operator p as the core, effectively overcoming the confusion between the concepts of operators and functions, which represents the limitation of traditional theory. In addition, the classical Laplace transform is integrated into our theory in a homomorphic form, revealing the principle that the Laplace transform is compatible with operational calculus theory.
This paper reports an interesting phenomenon in which fractional-order effects can be induced by the mismatch of the differential orders of space and time; that is, fractional-order effects can be ā¦
This paper reports an interesting phenomenon in which fractional-order effects can be induced by the mismatch of the differential orders of space and time; that is, fractional-order effects can be induced by spaceātime symmetry breakage. Classical mathematical equations can be transformed into differential equations of undetermined operators. We confirmed that the presence of fractional-order operator solutions in operator differential equations is contingent upon the mismatch of differential orders of space and time, which can induce both fractional operators in the time domain and fractional operators in the space domain. The introduction of symmetry breakage and operators of space and time offers novel insights into understanding nonlocal phenomena within the spaceātime continuum.
This paper reports an interesting phenomenon in which fractional-order effects can be induced by the mismatch of the differential orders of space and time; that is, fractional-order effects can be ā¦
This paper reports an interesting phenomenon in which fractional-order effects can be induced by the mismatch of the differential orders of space and time; that is, fractional-order effects can be induced by spaceātime symmetry breakage. Classical mathematical equations can be transformed into differential equations of undetermined operators. We confirmed that the presence of fractional-order operator solutions in operator differential equations is contingent upon the mismatch of differential orders of space and time, which can induce both fractional operators in the time domain and fractional operators in the space domain. The introduction of symmetry breakage and operators of space and time offers novel insights into understanding nonlocal phenomena within the spaceātime continuum.
This paper is dedicated to elucidating the abstract algebraic structure of operational calculus theory. Based on abstract algebra and operational calculus, the operator algebra theory of MikusiÅski has been revised. ā¦
This paper is dedicated to elucidating the abstract algebraic structure of operational calculus theory. Based on abstract algebra and operational calculus, the operator algebra theory of MikusiÅski has been revised. We restate the concept of MikusiÅskiās operator field as a convolutional field and define a new concept of operator field with the differential operator p as the core, effectively overcoming the confusion between the concepts of operators and functions, which represents the limitation of traditional theory. In addition, the classical Laplace transform is integrated into our theory in a homomorphic form, revealing the principle that the Laplace transform is compatible with operational calculus theory.
In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be ā¦
In recent years, our research on biomechanical and biophysical problems has involved a series of symmetry issues. We found that the fundamental laws of the aforementioned problems can all be characterized by fractal operators, and each type of operator possesses rich invariant properties. Based on the invariant properties of fractal operators, we discovered that the symmetry evolution laws of functional fractal trees in the physical fractal space can reveal the intrinsic correlations between special functions. This article explores the fractional-order correlation between special functions inspired by bone fractal operators. Specifically, the following contents are included: (1) showing the intrinsic expression in the convolutional kernel function of bone fractal operators and its correlation with special functions; (2) proving the following proposition: the convolutional kernel function of bone fractal operators is still related to the special functions under different input signals (external load, external stimulus); (3) using the bone fractal operators as the background and error function as the core, deriving the fractional-order correlation between different special functions.
In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we ā¦
In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we abstracted a family of fractal operators and investigate the kernel function and properties thereof. Based on fractal operators, the intrinsic relation between Bessel function and Struve function was revealed, and some new special functions were found. The results provide mathematical tools for biomechanics and automatic control.
This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between ā¦
This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: p+Ī±2-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators.
In this study, we delve into the general theory of operator kernel functions (OKFs) in operational calculus (OC). We established the rigorous mapping relation between the kernel function and the ā¦
In this study, we delve into the general theory of operator kernel functions (OKFs) in operational calculus (OC). We established the rigorous mapping relation between the kernel function and the corresponding operator through the primary translation operator eāpt, which bears a striking resemblance to the Laplace transform. Our research demonstrates the uniqueness of the kernel function, determined by the rules of operational calculus and its integral representation. This discovery provides a novel perspective on how the operational calculus can be understood and applied, particularly through convolution with kernel functions. We substantiate the accuracy of the proposed method by demonstrating the consistency between the operator solution and the classical solution for the heat conduction problem. Subsequently, on the fractal tree, fractal loop, and fractal ladder structures, we illustrate the application of operational calculus in viscoelastic constitutive and hemodynamics confirming that the method proposed unifies the OKFs in the existing OC theory and can be extended to the operator field. These results underscore the practical significance of our results and open up new possibilities for future research.
In this paper, we first consider the general fractional derivatives of arbitrary order defined in the RiemannāLiouville sense. In particular, we deduce an explicit form of their null space and ā¦
In this paper, we first consider the general fractional derivatives of arbitrary order defined in the RiemannāLiouville sense. In particular, we deduce an explicit form of their null space and prove the second fundamental theorem of fractional calculus that leads to a closed form formula for their projector operator. These results allow us to formulate the natural initial conditions for the fractional differential equations with the general fractional derivatives of arbitrary order in the RiemannāLiouville sense. In the second part of the paper, we develop an operational calculus of the MikusiÅski type for the general fractional derivatives of arbitrary order in the RiemannāLiouville sense and apply it for derivation of an explicit form of solutions to the Cauchy problems for the single- and multi-term linear fractional differential equations with these derivatives. The solutions are provided in form of the convolution series generated by the kernels of the corresponding general fractional integrals.
In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we ā¦
In this paper, we present a re-established functional fractal circuit model of arterial blood flow that incorporates the shunt effect of the branch vessels. Under the background of hemodynamics, we abstracted a family of fractal operators and investigate the kernel function and properties thereof. Based on fractal operators, the intrinsic relation between Bessel function and Struve function was revealed, and some new special functions were found. The results provide mathematical tools for biomechanics and automatic control.
MikusiÅski's operational calculus provides a way of applying the machinery of abstract algebra to the spaces and operators of calculus, thus allowing integro-differential equations to be solved by reducing them ā¦
MikusiÅski's operational calculus provides a way of applying the machinery of abstract algebra to the spaces and operators of calculus, thus allowing integro-differential equations to be solved by reducing them to algebraic equations. We summarise the application of this method to several operators of fractional calculus, defined by various convolution kernel functions at different levels of generality, and how the corresponding fractional differential equations can be solved.
Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate ā¦
Integral inequality plays a critical role in both theoretical and applied mathematics fields. It is clear that inequalities aim to develop different mathematical methods (numerically or analytically) and to dedicate the convergence and stability of the methods. Unfortunately, mathematical methods are useless if the method is not convergent or stable. Thus, there is a present day need for accurate inequalities in proving the existence and uniqueness of the mathematical methods. Convexity play a concrete role in the field of inequalities due to the behaviour of its definition. There is a strong relationship between convexity and symmetry. Which ever one we work on, we can apply to the other one due to the strong correlation produced between them especially in recent few years. In this article, we first introduced the notion of Ī» -incomplete gamma function. Using the new notation, we established a few inequalities of the HermiteāHadamard (HH) type involved the tempered fractional integrals for the convex functions which cover the previously published result such as Riemann integrals, RiemannāLiouville fractional integrals. Finally, three example are presented to demonstrate the application of our obtained inequalities on modified Bessel functions and q-digamma function.
A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested ā¦
A solution to the more than 300-years old problem of geometric and physical interpretation of fractional integration and differentiation (i.e., integration and differentiation of an arbitrary real order) is suggested for the Riemann-Liouville fractional integration and differentiation, the Caputo fractional differentiation, the Riesz potential, and the Feller potential. It is also generalized for giving a new geometric and physical interpretation of more general convolution integrals of the Volterra type. Besides this, a new physical interpretation is suggested for the Stieltjes integral.
(1949). Modern Operational Calculus for Undergraduates. The American Mathematical Monthly: Vol. 56, No. 5, pp. 295-300.
(1949). Modern Operational Calculus for Undergraduates. The American Mathematical Monthly: Vol. 56, No. 5, pp. 295-300.
The centenary of the birth of Oliver Heaviside last year has been the occasion of celebration by electrical engineers and physicists in this and other countries. In the discussions of ā¦
The centenary of the birth of Oliver Heaviside last year has been the occasion of celebration by electrical engineers and physicists in this and other countries. In the discussions of his work much has been said about the Operational Calculus ; and as the versions of its history which have been given both in these commemorative celebrations and in most of the textbooks of the subject are seriously incorrect, this may serve as an occasion to recount that history more correctly. The story in widest circulation is that the Operational Calculus was discovered by Heaviside (Boole being sometimesāand incorrectlyānamed as the discoverer of its applications to ordinary differential equations) and rejected by British mathematicians because of Heavisideās lack of rigour. The facts, as I shall show, are that the Calculus was well known in Britain and France before Heavisideās birth, and that the rejection of his paper had nothing to do with his use of symbolic methods.
Abstract In this paper, a stochastic space fractional advection diffusion equation of ItĆ“ type with one-dimensional white noise process is presented. The fractional derivative is defined in the sense of ā¦
Abstract In this paper, a stochastic space fractional advection diffusion equation of ItƓ type with one-dimensional white noise process is presented. The fractional derivative is defined in the sense of Caputo. A stochastic compact finite difference method is used to study the proposed model numerically. Stability analysis and consistency for the stochastic compact finite difference scheme are proved. Two test examples are given to test the performance of the proposed method. Numerical simulations show that the results obtained are compatible with the exact solutions and with the solutions derived in the literature.
Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread ā¦
Often referred to as special functions or mathematical functions, the origin of many members of the remarkably vast family of higher transcendental functions can be traced back to such widespread areas as (for example) mathematical physics, analytic number theory and applied mathematical sciences. Here, in this survey-cum-expository review article, we aim at presenting a brief introductory overview and survey of some of the recent developments in the theory of several extensively studied higher transcendental functions and their potential applications. For further reading and researching by those who are interested in pursuing this subject, we have chosen to provide references to various useful monographs and textbooks on the theory and applications of higher transcendental functions. Some operators of fractional calculus, which are associated with higher transcendental functions, together with their applications, have also been considered. Many of the higher transcendental functions, especially those of the hypergeometric type, which we have investigated in this survey-cum-expository review article, are known to display a kind of symmetry in the sense that they remain invariant when the order of the numerator parameters or when the order of the denominator parameters is arbitrarily changed.
This essay-review of Jeffreysā very welcome and valuable Tract with the above title has been written at the editorās request. Many readers of the Gazette must have heard of Heavisideās ā¦
This essay-review of Jeffreysā very welcome and valuable Tract with the above title has been written at the editorās request. Many readers of the Gazette must have heard of Heavisideās operational method of solving the equations of dynamics and mathematical physics. If they have tried to learn about them from Heavisideās own works, they have attempted a difficult task. Nothing more obscure than his mathematical writings is known to me. A Cambridge Tract is now at their disposal. Prom it much may be learned; but the air of mystery stillāat least in partāremains.
This book is designed to be read by students who have had a first elementary course in general algebra. It provides a common generalization of the primes of arithmetic and ā¦
This book is designed to be read by students who have had a first elementary course in general algebra. It provides a common generalization of the primes of arithmetic and the points of geometry. The book explains the various elementary operations which can be performed on ideals.
The properties of the unique nontrivial analytic solution, defined implicitly by a functional equation, are pointed out. This work provides local estimations and global inequalities for the involved solution. The ā¦
The properties of the unique nontrivial analytic solution, defined implicitly by a functional equation, are pointed out. This work provides local estimations and global inequalities for the involved solution. The corresponding operatorial equation is studied as well. The second part of the paper is devoted to the full classical moment problem, which is an inverse problem. Two constraints are imposed on the solution. One of them requires the solution to be dominated by a concrete convex operator defined on the positive cone of the domain space. A one-dimensional operator is valued, and a multidimensional scalar moment problem is solved. In both cases, the existence and the uniqueness of the solution are proved. The general idea of the paper is to provide detailed information on solutions which are not expressible in terms of elementary functions.
This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between ā¦
This article studies the error function and its invariance properties in the convolutional kernel function of bone fractal operators. Specifically, the following contents are included: (1) demonstrating the correlation between the convolution kernel function and error function of bone fractal operators; (2) focusing on the main part of bone fractal operators: p+Ī±2-type differential operator, discussing the convolutional kernel function image; (3) exploring the fractional-order correlation between the error function and other special functions from the perspective of fractal operators.
Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent ā¦
Geometric function theory has extensively explored the geometric characteristics of analytic functions within symmetric domains. This study analyzes the geometric properties of a specific class of analytic functions employing confluent hypergeometric functions and generalized Bessel functions of the first kind. Specific constraints are imposed on the parameters to ensure the inclusion of the confluent hypergeometric function within the analytic function class. The coefficient bound of the class is used to determine the inclusion properties of integral operators involving generalized Bessel functions of the first kind. Different results are observed for these operators, depending on the specific values of the parameters. The results presented here include some previously published findings as special cases.
The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore ā¦
The interest in special complex functions and their wide-ranging implementations in geometric function theory (GFT) has developed tremendously. Recently, subordination theory has been instrumentally employed for special functions to explore their geometric properties. In this effort, by using a convolutional structure, we combine the geometric series, logarithm, and HurwitzāLerch zeta functions to formulate a new special function, namely, the logarithm-HurwitzāLerch zeta function (LHL-Z function). This investigation then contributes to the study of the LHL-Z function in terms of the geometric theory of holomorphic functions, based on the differential subordination methodology, to discuss and determine the univalence and convexity conditions of the LHL-Z function. Moreover, there are other subordination and superordination connections that may be visually represented using geometric methods. Functions often exhibit symmetry when subjected to conformal mappings. The investigation of the symmetries of these mappings may provide a clearer understanding of how subordination and superordination with the HurwitzāLerch zeta function behave under different transformations.
This paper is dedicated to elucidating the abstract algebraic structure of operational calculus theory. Based on abstract algebra and operational calculus, the operator algebra theory of MikusiÅski has been revised. ā¦
This paper is dedicated to elucidating the abstract algebraic structure of operational calculus theory. Based on abstract algebra and operational calculus, the operator algebra theory of MikusiÅski has been revised. We restate the concept of MikusiÅskiās operator field as a convolutional field and define a new concept of operator field with the differential operator p as the core, effectively overcoming the confusion between the concepts of operators and functions, which represents the limitation of traditional theory. In addition, the classical Laplace transform is integrated into our theory in a homomorphic form, revealing the principle that the Laplace transform is compatible with operational calculus theory.
Introduction. In the present paper an calculus for functions of n variables is developed, analogous to that of Mikusinski for functions of one variable [6], and applications of this calculus ā¦
Introduction. In the present paper an calculus for functions of n variables is developed, analogous to that of Mikusinski for functions of one variable [6], and applications of this calculus to linear partial differential equations with constant coefficients are given. As is known (see [13]), the Laplace transform method (and other transform methods) may be used to reduce partial differential equation to an ordinary differential equation. The advantage of Mikusinski's calculus over the Laplace transform method (besides being well justified) lies in the fact that it can be applied in cases in which Laplace transform does not exist. To apply his calculus in partial differential equations Mikusinski introduces the notion of an operational function1 and reduces the solution of partial differential equation in two variables to solution of an ordinary differential equation. Using generalized functions [3], [4] one usually obtains theorems about existence and uniqueness of solutions of partial differential equations without being able to find them explicitly. The calculus developed here has the same advantage over the Laplace transform method as Mikusinski's calculus. Although closely related to Mikusinski's calculus, it seems to us to comprise, in comparison with the latter, further step towards the algebraization of solutions of partial differential equations. This is achieved by omitting the notion of the operational used by Mikusinski and by developing and justifying purely algebraic method of solutions of partial differential equations. The paper consists of four sections. In ? 1, the basic definitions and theorems of the calculus are given. Except for the definitions of some new operators and theorems concerning these operators, the material of? 1 is partially known and is given here only for the sake of completeness. However we stress that the notion of an as introduced in this work depends on some region in n-dimensional Euclidean space Rn. In ? 2 sufficient conditions are found for an to be function (Theorem 1), and the notion of a function contained in an operator is introduced. The existence of such function and its uniqueness for some operators are also shown (Theorem 4). Similar to the notion of an operator, the notion of function contained in an also depends on the region in Rn.
A Ffactional element model describes a special kind of viscoelastic material. Its stress is proportional to the fractional-order derivative of strain. Physically the mechanical analogies of fractional elements can be ā¦
A Ffactional element model describes a special kind of viscoelastic material. Its stress is proportional to the fractional-order derivative of strain. Physically the mechanical analogies of fractional elements can be represented by spring-dashpot fractal networks. We introduce a constitutive operator in the constitutive equations of viscoelastic materials. To derive constitutive operators for spring-dashpot fractal networks, we use Heaviside operational calculus, which provides explicit answers not otherwise obtainable simply. Then the series-parallel formulas for the constitutive operator are derived. Using these formulas, a constitutive equation of fractional element with 1/2-order derivative is obtained. Finally we find the way to derive the constitutive equations with other fractional-order derivatives and their mechanical analogies.