Some Parseval-Goldstein Type Theorems For Generalized Integral Transforms

In this work, we establish some Parseval-Goldstein type identities and relations that include various new generalized integral transforms such as $\mathcal{L}_{\alpha,\mu}$-transform and generalized Stieltjes transform. In addition, we evaluated improper integrals of some fundamental and special functions using our results

Axisymmetric Stokes flow due to a point-force singularity acting between two coaxially positioned rigid no-slip disks

We investigate theoretically on the basis of the steady Stokes equations for a viscous incompressible fluid the flow induced by a Stokeslet located on the centre axis of two coaxially positioned rigid disks. The Stokeslet is directed along the centre axis. No-slip boundary conditions are assumed to hold at the surfaces of the disks. We perform the calculation of the associated Green's function in large parts analytically, reducing the spatial evaluation of the flow field to one-dimensional integrations amenable to numerical treatment. To this end, we formulate the solution of the hydrodynamic problem for the viscous flow surrounding the two disks as a mixed-boundary-value problem, which we then reduce into a system of four dual integral equations. We show the existence of viscous toroidal eddies arising in the fluid domain bounded by the two disks, manifested in the plane containing the centre axis through adjacent counterrotating eddies. Additionally, we probe the effect of the confining disks on the slow dynamics of a point-like particle by evaluating the hydrodynamic mobility function associated with axial motion. Thereupon, we assess the appropriateness of the commonly-employed superposition approximation and discuss its validity and applicability as a function of the geometrical properties of the system. Additionally, we complement our semi-analytical approach by finite-element computer simulations, which reveals a good agreement. Our results may find applications in guiding the design of microparticle-based sensing devices and electrokinetic transport in small scale capacitors

A study of discrete and integral transforms with 3 logarithmic separable kernels

In this thesis, we will be examining different classes of discrete and integral transforms. We start with a general class of integral transforms which include those logarithmic separable kernels. Transforms with logarithmic separable kernels include the Fourier transform, the Laplace transform and the Mellin transform. The shifting and convolution properties for this class of transforms are examined, and sufficient conditions which guarantee the existence of the convolution formula will be given. It will be shown that a subclass of these integral operators are injective and an inversion formula will be presented on some class of continuously differentiable functions. We will apply these results to second-order differential equations to obtain new analytical solutions to these equations and compare these to a numerical solution

Revisiting the Anisotropic Fractional Calder\'on Problem Using the Caffarelli-Silvestre Extension

We revisit the source-to-solution anisotropic fractional Calder\'on problem introduced and analyzed in [FGKU21] and [F21]. Using the Caffarelli-Silvestre interpretation of the fractional Laplacian, we provide an alternative argument for the recovery of the heat and wave kernels from [FGKU21]. This shows that in the setting of the source-to-solution anisotropic fractional Calder\'on problem the heat and Caffarelli-Silvestre approach give rise to equivalent perspectives and that each kernel can be recovered from the other. Moreover, we also discuss the Dirichlet-to-Neumann anisotropic source-to-solution problem and provide a direct link between the Dirichlet Poisson kernel and the wave kernel. This illustrates that it is also possible to argue completely on the level of the Poisson kernel, bypassing the recovery of the heat kernel as an additional auxiliary step. Last but not least, as in [CGRU23], we relate the local and nonlocal source-to-solution Calder\'on problems.Comment: 26 pages, comments welcom