General Fractional Vector Calculus

A generalization of fractional vector calculus (FVC) as a self-consistent mathematical theory is proposed to take into account a general form of non-locality in kernels of fractional vector differential and integral operators. Self-consistency involves proving generalizations of all fundamental theorems of vector calculus for generalized kernels of operators. In the generalization of FVC from power-law nonlocality to the general form of nonlocality in space, we use the general fractional calculus (GFC) in the Luchko approach, which was published in 2021. This paper proposed the following: (I) Self-consistent definitions of general fractional differential vector operators: the regional and line general fractional gradients, the regional and surface general fractional curl operators, the general fractional divergence are proposed. (II) Self-consistent definitions of general fractional integral vector operators: the general fractional circulation, general fractional flux and general fractional volume integral are proposed. (III) The general fractional gradient, Green’s, Stokes’ and Gauss’s theorems as fundamental theorems of general fractional vector calculus are proved for simple and complex regions. The fundamental theorems (Gradient, Green, Stokes, Gauss theorems) of the proposed general FVC are proved for a wider class of domains, surfaces and curves. All these three parts allow us to state that we proposed a calculus, which is a general fractional vector calculus (General FVC). The difficulties and problems of defining general fractional integral and differential vector operators are discussed to the nonlocal case, caused by the violation of standard product rule (Leibniz rule), chain rule (rule of differentiation of function composition) and semigroup property. General FVC for orthogonal curvilinear coordinates, which includes general fractional vector operators for the spherical and cylindrical coordinates, is also proposed.

Fractional solutions for the inextensible Heisenberg antiferromagnetic fow and solitonic magnetic flux surfaces in the binormal direction

Motivated by recent researches in magnetic curves and their flows in different types of geometric manifolds and physical spacetime structures, we compute Lorentz force equations associated with the magnetic b-lines in the binormal direction. Evolution equations of magnetic b-lines due to inextensible Heisenberg antiferromagnetic flow are computed to construct the soliton surface associated with the inextensible Heisenberg antiferromagnetic flow. Then, their explicit solutions are investigated in terms of magnetic and geometric quantities via the conformable fractional derivative method. By considering arc-length and time-dependent orthogonal curvilinear coordinates, we finally determine the necessary and sufficient conditions that have to be satisfied by these quantities to define the Lorentz magnetic flux surfaces based on the inextensible Heisenberg antiferromagnetic flow model.

Stresses and deformations of an eccentric cylindrical cam during hydro-joining

The stresses and deformations of an eccentric cylindrical cam during the process of hydro-joining have been calculated in orthogonal curvilinear coordinates according to the mechanical model of the cam and governing equations in terms of appropriate complex potentials with suitable boundary conditions. The radial and shearing stress coefficients determined for an example cam are far less than that of the tangential stress during hydro-joining. The tangential stress coefficient of the outer surface of the cam is greater than that of the internal surface when the polar angle exceeds a particular value. The position of the maximum value of the radial stress coefficient is located on the internal surface of the cam, and the maximum shear stress coefficient is located between the inside and outside surfaces of the cam. The cam deformations on the internal and external surfaces under internal pressure respectively attain maximum values at particular angles. The maximum values of the radial and y-directional deformations are located at the position of the minimum wall thickness. The radial deformations determined for an example cam are far larger that the tangential deformations during hydro-joining. The errors between the theoretical and numerical solutions for the tangential stress and the y-directional deformation are both very small.