On weighted Adams-Bashforth rules

One class of the linear multistep methods for solving the Cauchy problems of the form $y\u27=F(x,y)$, $y(x_{0})=y_{0}$, contains Adams-Bashforth rules of the form $y_{n+1}=y_{n}+hsum_{i=0}^{k-1} B_i^{(k)} F(x_{n-i},y_{n-i})$, where ${ B_i^{(k)}} _{i = 0}^{k - 1}$ are fixed numbers. In this paper, we propose an idea for weighted type of Adams-Bashforth rules for solving the Cauchy problem for singular differential equations,[A(x)y\u27+B(x)y=G(x,y), quad y(x_0)=y_0,]where $A$ and $B$ are two polynomials determining the well-known classical weight functions in the theory of orthogonal polynomials. Some numerical examples are also included

On weighted Adams-Bashforth rules

One class of the linear multistep methods for solving the Cauchy problems of the form $y\u27=F(x,y)$, $y(x_{0})=y_{0}$, contains Adams-Bashforth rules of the form $y_{n+1}=y_{n}+hsum_{i=0}^{k-1} B_i^{(k)} F(x_{n-i},y_{n-i})$, where ${ B_i^{(k)}} _{i = 0}^{k - 1}$ are fixed numbers. In this paper, we propose an idea for weighted type of Adams-Bashforth rules for solving the Cauchy problem for singular differential equations,[A(x)y\u27+B(x)y=G(x,y), quad y(x_0)=y_0,]where $A$ and $B$ are two polynomials determining the well-known classical weight functions in the theory of orthogonal polynomials. Some numerical examples are also included

A numerical method for solving two-dimensional nonlinear parabolic problems based on a preconditioning operator

‎This article considers a nonlinear system of elliptic problems, which is obtained by discretizing the time variable of a two-dimensional nonlinear parabolic problem. Since the system consists of ill-conditioned problems, therefore a stabilized, mesh-free method is proposed. The method is based on coupling the preconditioned Sobolev space gradient method and WEB-spline finite element method with Helmholtz operator as a preconditioner. The convergence and error analysis of the method are given. Finally, a numerical example is solved by this preconditioner to show the efficiency and accuracy of the proposed methods

Inverse source problem in a space fractional diffusion equation from the final overdetermination

summary:We consider the problem of determining the unknown source term $f=f(x,t)$ in a space fractional diffusion equation from the measured data at the final time $u(x,T) = \psi (x)$. In this way, a methodology involving minimization of the cost functional $J(f) = \int _0^l ( u(x,t;f )|_{t = T} - \psi (x)) ^2 {\rm d}x$ is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and convergence of the sequence $\{J'(f^{(n)}) \}$, where $f^{(n)}$ is the $n$th iteration of a gradient like method. At the end, the convexity of the Fréchet derivative is given

ZnO, Cu-doped ZnO, Al-doped ZnO and Cu-Al doped ZnO thin films: Advanced micro-morphology, crystalline structures and optical properties

The thin film coatings composed of: undoped ZnO film, ZnO doped with Al, ZnO doped with Cu, and ZnO simultaneously doped with Al and Cu (co-doping) were separately deposited on quartz substrates using RF sputtering method with different targets. The advanced fractal features, crystalline structure and optical properties of sputtered samples were investigated by atomic force microscopy (AFM), X‐ray diffraction (XRD) and UV–vis spectroscopy. Microstructural studies revealed homogeneously granular structure of ZnO layer and axially oriented granular structure of AZO thin film.The transmission spectra of undoped, mono-doped and co-doped ZnO thin films were measured revealing relatively large transmittance of more than 80 % for un-doped and co-doped samples and less than that value for mono-doped thin films in both visible and infrared regions. CAZO thin film was found the most transparent thin film in the visible area being a prerequisite for good TCO. Analysis of absorption coefficients demonstrated that excitonic effects are invisible in mono-doped and co-doped samples. Also, PL spectra show that in these samples there are very high densities of free carriers and presence of impurities, which is important for conductivity of thin films as well as their optical applications. The optical band gap of ZnO thin films decreases by Cu doping from 3.12 eV to 3.09 eV and increases by Al doping to 4.30 eV, but remains exactly between those values in terms of co-doped sample (3.75 eV)