Approximation Theorems Associated with Multidimensional Fractional Fourier Transform and Applications in Laplace and Heat Equations

Abstract

In this paper, we establish two approximation theorems for the multidimensional fractional Fourier transform via appropriate convolutions. As applications, we study the boundary and initial problems of the Laplace and heat equations with chirp functions. Furthermore, we obtain the general Heisenberg inequality with respect to the multidimensional fractional Fourier transform.

A Singular Tempered Sub-Diffusion Fractional Model Involving a Non-Symmetrically Quasi-Homogeneous Operator

“…From Relation (2), it is easy to see that the tempered fractional derivative becomes the Riemann-Liouville fractional derivative, provided that µ = 0. This suggests that the tempered fractional derivative extends beyond classical fractional calculus, encompassing special cases like the Riemann-Liouville derivatives, Caputo fractional derivatives, and others [9][10][11][12][13][14][15]. Recently, by employing Ricceri's variational principle, Ledesma et al [16] studied a tempered fractional sub-diffusion model with an oscillating term:…”

A Singular Tempered Sub-Diffusion Fractional Equation with Changing-Sign Perturbation

“…In [21], Zakarya et al provided novel generalizations by considering the generalized conformable fractional integrals for reverse Copson's type inequalities on time scales. For some other applications of fractional calculus, the reader is referred to [22][23][24][25][26][27][28][29][30][31][32][33][34][35][36]. This paper deals with the following sub-diffusion model with a changing-sign perturbation.…”

The Iterative Properties for Positive Solutions of a Tempered Fractional Equation

“…Since fractional order differential equations can describe many natural phenomena with long-time behavior such as abnormal dispersion, analytical chemistry, biological sciences, artificial neural network, time-frequency analysis, and so on, the theories of fractional calculus have attracted the attention of a large number of mathematical researchers [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. The classical fractional integrals and derivatives are only convolutions by using a power law, such as the Riemann-Liouville fractional derivatives [21][22][23], the Caputo fractional derivatives [24,25], the Hilfer fractional derivative [26], the Atangana-Baleanu-Caputo fractional derivative [27], Hadamard fractional derivatives [28,29], and so on, which fail to model the limits of random walk if they have an exponentially tempered jump distribution [30] exhibiting the semi-heavy tails or semi-long range dependence.…”