Fundamental Solutions to Time-Fractional Advection Diffusion Equation in a Case of Two Space Variables

The fundamental solutions to time-fractional advection diffusion equation in a plane and a half-plane are obtained using the Laplace integral transform with respect to time t and the Fourier transforms with respect to the space coordinates x and y. The Cauchy, source, and Dirichlet problems are investigated. The solutions are expressed in terms of integrals of Bessel functions combined with Mittag-Leffler functions. Numerical results are illustrated graphically

Reakcija β-amino-α,γ-dicianokrotononitrila s acetofenonom: sinteza derivata piridina, piridazina i tiofena s antimikrobnim djelovanjem

Condensation of β-amino-α,γ-dicyanocrotononitrile (1) with acetophenone gave the 2-amino-4-phenylpenta-1,3-diene-1,1,3-tricarbonitrile (2). The latter product was used in a series of heterocyclization reactions when react with different reagents like diazonium salts, hydrazines, hydroxylamine and elemental sulfur to give pyridazine, pyrazole, isoxazole and thiophene derivatives, respectively. On the other hand, it gave pyridine derivatives with aromatic aldehydes followed by reaction with cyanomethylene reagents. The MIC values for the newly synthesized product were measured against E. coli, B. cereus, B. subtilis and C. albicansKondenzacijom β-amino-α,γ-dicijanokrotononitrila 1 s acetofenonom dobiven je 2-amino-4-fenilpenta-1,3-dien-1,1,3-trikarbonitril (2) koji je upotrebljen u reakcijama heterociklizacije s različitim reagensima poput diazonijevih soli, hidrazina, hidroksilamina i elementarnog sumpora pri čemu su nastali derivati piridazina, pirazola, izoksazola, odnosno tiofena. Spoj 2 je u reakciji s aromatskim aldehidima te naknadno sa cijanometilenima dao derivate piridina. Određene su MIC vrijednosti za novosintetizirane spojeve protiv E. coli, B. cereus, B. subtilis i C. albicans

Solutions to Time-Fractional Diffusion-Wave Equation in Spherical Coordinates

Solutions to time-fractional diffusion-wave equation with a source term in spherical coordinates are obtained for an infinite medium. The solutions are found using the Laplace transform with respect to time t, the finite Fourier transform with respect to the angular coordinate , the Legendre transform with respect to the spatial coordinate , and the Hankel transform of the order n+1/2 with respect to the radial coordinate . In the central symmetric case with one spatial coordinate the obtained results coincide with those studied earlier

Solutions to fractional diffusion-wave equation in a circular sector

The time-fractional diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a domain 0 ≤ r < R, 0 < ϕ < ϕ0 under different boundary conditions. The Laplace integral transform with respect to time, the finite Fourier transforms with respect to the angular coordinate, and the finite Hankel transforms with respect to the radial coordinate are used. The fundamental solutions are expressed in terms of the Mittag-Leffler function. The particular cases of the obtained solutions corresponding to the diffusion equation (α = 1) and the wave equation (α = 2) coincide with those known in the literature

Axisymmetric solutions to fractional diffusion-wave equation in a cylinder under Robin boundary condition

The axisymmetric time-fractional diffusion-wave equation with the Caputo derivative of the order 0 < α ≤ 2 is considered in a cylinder under the prescribed linear combination of the values of the sought function and the values of its normal derivative at the boundary. The fundamental solutions to the Cauchy, source, and boundary problems are investigated. The Laplace transform with respect to time and finite Hankel transform with respect to the radial coordinate are used. The solutions are obtained in terms of Mittag-Leffler functions. The numerical results are illustrated graphically

Axisymmetric solutions to the Cauchy problem for time-fractional diffusion equation in a circle

The Cauchy problems for time-fractional diffusion equation with delta pulse initial value of a sought-for function is studied in a circle domain in the axisymmetric case under zero Dirichlet and Neumann boundary conditions, respectively. The Caputo fractional derivative is used. The Laplace and finite Hankel integral transforms are employed. The results are illustrated graphically