Ashyralyev, Allaberen; Cakir, Zafer; Lukashov, A.; Ashyralyev, A.

In this work, we investigate initial-boundary value problems for fractional parabolic equations with the Neumann boundary condition. Stability estimates for the solution of this problem are established. Difference schemes for approximate solution of initial-boundary value problem are constructed. Furthermore, we give theorem on coercive stability estimates for the solution of the difference schemes.

Ashyralyev, Allaberen; Emirov, Nazar; Cakir, Zafer

We study initial-boundary value problems for fractional parabolic equations with the Dirichlet-Neumann conditions. We obtain a stable difference schemes for this problem, and obtain theorems on coercive stability estimates for the solution of the first order of accuracy difference scheme. A procedure of modified Gauss elimination method is applied for the solution of the first and second order of accuracy difference schemes of one-dimensional fractional parabolic differential equations.

Ashyralyev, Allaberen; Cakir, Zafer

The second order of accuracy stable difference scheme for the numerical solution of the mixed problem for the fractional parabolic equation are presented using by r-modified Crank-Nicholson difference scheme. Stability estimate for the solution of this difference scheme is obtained. A procedure of modified Gauss elimination method is used for solving this difference scheme in the case of one-dimensional fractional parabolic partial differential equations. Numerical results for this scheme and the Crank-Nicholson scheme are compared in test examples.

Ashyralyyev, Charyyar; Cakir, Zafer; Ashyralyev, A; Malkowsky, E

In this paper, some sequences of matrices for matrix pencils are considered. Properties of small singular values of these matrices are investigated. Estimates for singular values are obtained. The result of computer calculations is given.

Ashyralyyev, Charyyar; Cakir, Zafer; Ashyralyev, A; Malkowsky, E

Approximate quadrature formulas for the numerical calculation of the two-dimensional Vekua potential and singular integrals are obtained. The mechanical quadrature method for the two-dimensional quasilinear singular integral equation with Vekua operators is described. The numerical results are compared with the exact solution of the integral equation.

Ashyralyev, Allaberen; Artykov, Merdan; Cakir, Zafer; Ashyralyev, A; Malkowsky, E

In this work, the first and second order of difference schemes that approximately solve the initial-boundary value problem for multidimensional fractional parabolic equation with Dirichlet boundary condition are presented. The stability estimates for the solution of these difference schemes are established.

Cakir, Zafer

Following Grossman and Katz (Non-Newtonian Calculus, 1972), we construct the sets B(A) and C(A) of geometric complex-valued bounded and continuous functions, where A denotes the compact subset of the complex plane C. We show that the sets B(A) and C(A) of complex-valued bounded and continuous functions form a vector space with respect to the addition and scalar multiplication in the sense of multiplicative calculus. Finally, we prove that B(A) and C(A) are complete metric spaces.

Ashyralyev, Allaberen; Cakir, Zafer

In the present study, the first and second order of accuracy stable difference schemes for the numerical solution of the initial boundary value problem for the fractional parabolic equation with the Neumann boundary condition are presented. Almost coercive stability estimates for the solution of these difference schemes are obtained. The method is illustrated by numerical examples.

Ashyralyev, Allaberen; Cakir, Zafer

The first and second order of accuracy stable difference schemes for the numerical solution of the mixed problem for the fractional parabolic equation are presented. Stability and almost coercive stability estimates for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes in the case of one-dimensional fractional parabolic partial differential equations.

Cakir, Zafer

The stable difference schemes for the fractional parabolic equation with Dirichlet and Neumann boundary conditions are presented. Stability estimates and almost coercive stability estimates with ln (1 / ( + | h |)) for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes of one-dimensional fractional parabolic partial differential equations.

Ashyralyev, Allaberen; Emirov, Nazar; Cakir, Zafer; Ashyralyev, A; Lukashov, A

The multidimensional fractional parabolic equation with the Dirichlet-Neumann condition is studied. Stability estimates for the solution of the initial-boundary value problem for this fractional parabolic equation are established. The stable difference schemes for this problem are presented. Stability estimates for the solution of the first order of accuracy difference scheme are obtained. A procedure of modified Gauss elimination method is applied for the solution of first and second order of accuracy difference schemes of one-dimensional fractional parabolic differential equations.

Ashyralyev, Allaberen; Cakir, Zafer; Simos, TE

The stable difference schemes for the multidimensional fractional parabolic equation with Dirichlet and Neumann boundary conditions are presented. Stability estimates and almost coercive stability estimates with ln 1/τ+|h| for the solution of these difference schemes are obtained. A procedure of modified Gauss elimination method is used for solving these difference schemes of one-dimensional fractional parabolic partial differential equations.