scholarly journals Product Rule and Chain Rule Estimates for Fractional Derivatives on Spaces that Satisfy the Doubling Condition

2002 ◽  
Vol 188 (1) ◽  
pp. 27-37 ◽  
Author(s):  
A.Eduardo Gatto
Keyword(s):  
Fractional Derivatives ◽  
Chain Rule ◽  
Product Rule ◽  
Doubling Condition

scholarly journals Product rule for vector fractional derivatives

2012 ◽  
Vol 15 (3) ◽  
Author(s):  
Diogo Bolster ◽  
Mark Meerschaert ◽  
Alla Sikorskii
Keyword(s):  
Vector Space ◽  
Fractional Derivatives ◽  
Fourier Transforms ◽  
Product Rule ◽  
Finite Dimensional ◽  
Euclidean Vector Space ◽  
Derivatives Of

AbstractThis paper establishes a product rule for fractional derivatives of a realvalued function defined on a finite dimensional Euclidean vector space. The proof uses Fourier transforms.

scholarly journals The Triple Product Rule is Incorrect

2021 ◽  
Vol 1 (2) ◽  
pp. 1-3
Author(s):  
Igor Stepanov*
Keyword(s):  
Heat Capacity ◽  
Thermal Expansion ◽  
Isothermal Compressibility ◽  
Chain Rule ◽  
Triple Product ◽  
Product Rule ◽  
Capacity Ratio ◽  
The Difference ◽  
The Relationship ◽  
Derivatives Of

The triple product rule, also known as the cyclic chain rule, cyclic relation, cyclical rule or Euler's chain rule, relates the partial derivatives of three interdependent variables, and often finds application in thermodynamics. It is shown here that its derivation is wrong, and that this rule is not correct; hence, the Mayer's relation and the heat capacity ratio, which describe the difference between isobaric and isochoric heat capacities, are also untrue. Also, the relationship linking thermal expansion and isothermal compressibility is wrong. These results are confirmed by many experiments and by the previous theoretical findings of the author.

Mathematics Refresher

2021 ◽  
pp. 183-186
Author(s):  
Timothy E. Essington
Keyword(s):  
Chain Rule ◽  
Identity Matrix ◽  
Product Rule ◽  
Sum Rule ◽  
Matrix Operations

The chapter “Mathematics Refresher” provides a brief reminder of operations with logarithms, matrices, and calculus, for student reference. It starts off by reviewing the differences between regular logarithms and natural logarithms and provides some examples of common operations with logarithms. It then introduces derivatives and integrals (although it is never necessary to compute an integral in this book, it is still useful to know what an integral is) and explains the sum rule, the product rule, the quotient rule, and the chain rule. Next, it provides a brief overview of matrices and matrix operations, including matrix dimensions, and addition and multiplication of matrices. It concludes with a discussion of the identity matrix.

scholarly journals The Triple Product Rule is Incorrect

2021 ◽  
pp. 1-3
Author(s):  
Igor Stepanov ◽  
Keyword(s):  
Heat Capacity ◽  
Thermal Expansion ◽  
Isothermal Compressibility ◽  
Chain Rule ◽  
Triple Product ◽  
Product Rule ◽  
Capacity Ratio ◽  
The Difference ◽  
The Relationship ◽  
Derivatives Of

The triple product rule, also known as the cyclic chain rule, cyclic relation, cyclical rule or Euler’s chain rule, relates the partial derivatives of three interdependent variables, and often finds application in thermodynamics. It is shown here that its derivation is wrong, and that this rule is not correct; hence, the Mayer’s relation and the heat capacity ratio, which describe the difference between isobaric and isochoric heat capacities, are also untrue. Also, the relationship linking thermal expansion and isothermal compressibility is wrong. These results are confirmed by many experiments and by the previous theoretical findings of the author.

scholarly journals Some types of first order fractional differential equations and their applications

2021 ◽  
Vol 268 ◽  
pp. 01073
Author(s):  
Chiihuei Yu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Chain Rule ◽  
The Other ◽  
General Solutions ◽  
First Order ◽  
Other Hand ◽  
Fractional Differential

This paper uses a new multiplication of fractional functions and chain rule for fractional derivatives, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives to obtain the general solutions of four types of first order fractional differential equations. On the other hand, some examples are proposed to illustrate our results.

scholarly journals Summation Formulas Obtained by Means of the Generalized Chain Rule for Fractional Derivatives

2014 ◽  
Vol 2014 ◽  
pp. 1-7 ◽  
Author(s):  
S. Gaboury ◽  
R. Tremblay
Keyword(s):  
General Formula ◽  
Hypergeometric Functions ◽  
Fractional Derivatives ◽  
Special Functions ◽  
Mathematical Physics ◽  
Chain Rule ◽  
Generalized Hypergeometric Functions ◽  
Summation Formulas ◽  
Several Variables ◽  
Special Cases

In 1970, several interesting new summation formulas were obtained by using a generalized chain rule for fractional derivatives. The main object of this paper is to obtain a presumably new general formula. Many special cases involving special functions of mathematical physics such as the generalized hypergeometric functions, the Appell F1 function, and the Lauricella functions of several variables FD(n) are given.

scholarly journals Research on some types of fractional differential equations which can be transformed into separable variables

2021 ◽  
Vol 268 ◽  
pp. 01080
Author(s):  
Chiihuei Yu
Keyword(s):  
Differential Equations ◽  
Fractional Differential Equations ◽  
Fractional Derivatives ◽  
Product Rule ◽  
Fractional Differential

In this paper, we study some types of fractional differential equations which can be transformed into separable variables, regarding the Jumarie type of modified Riemann-Liouville fractional derivatives. We use a new multiplication of fractional functions and product rule for fractional derivatives to obtain the solutions of these fractional differential equations. Furthermore, some examples are given to demonstrate our results.

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