High-order compact solution of the one-dimensional heat and advection–diffusion equations

In this work, we propose a high-order accurate method for solving the one-dimensional heat and advection–diffusion equations. We apply a compact finite difference approximation of fourth-order for discretizing spatial derivatives of these equations and the cubic C¹$C^1$-spline collocation method for the resulting linear system of ordinary differential equations. The cubic C¹$C^1$-spline collocation method is an A-stable method for time integration of parabolic equations. The proposed method has fourth-order accuracy in both space and time variables, i.e. this method is of order O(h⁴,k⁴)$O(h^4\text{,}{k}^4)$. Additional to high-order of accuracy, the proposed method is unconditionally stable which will be proved in this paper. Numerical results show that the compact finite difference approximation of fourth-order and the cubic C¹$C^1$-spline collocation method give an efficient method for solving the one-dimensional heat and advection–diffusion equations.     

Solving nonlinear fractional partial differential equations using the homotopy analysis method

In this article, the homotopy analysis method is applied to solve nonlinear fractional partial differential equations. On the basis of the homotopy analysis method, a scheme is developed to obtain the approximate solution of the fractional KdV, K(2,2), Burgers, BBM‐Burgers, cubic Boussinesq, coupled KdV, and Boussinesq‐like B(m,n) equations with initial conditions, which are introduced by replacing some integer‐order time derivatives by fractional derivatives. The homotopy analysis method for partial differential equations of integer‐order is directly extended to derive explicit and numerical solutions of the fractional partial differential equations. The solutions of the studied models are calculated in the form of convergent series with easily computable components. The results of applying this procedure to the studied cases show the high accuracy and efficiency of the new technique. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Numerical solution of hyperbolic telegraph equation using the Chebyshev tau method

In this article we propose a numerical scheme to solve the one‐dimensional hyperbolic telegraph equation. The method consists of expanding the required approximate solution as the elements of shifted Chebyshev polynomials. Using the operational matrices of integral and derivative, we reduce the problem to a set of linear algebraic equations. Some numerical examples are included to demonstrate the validity and applicability of the technique. The method is easy to implement and produces very accurate results. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Solution of a laminar boundary layer flow via a numerical method

In this paper, the numerical solution of the Blasius problem is obtained using the collocation method based on rational Chebyshev functions. The Blasius equation is a nonlinear ordinary differential equation which arises in the boundary layer flow. The method reduces solving the equation to solving a system of nonlinear algebraic equations. The results presented here demonstrate reliability and efficiency of the method.     

Rational Legendre pseudospectral approach for solving nonlinear differential equations of Lane–Emden type

Lane–Emden equation is a nonlinear singular equation in the astrophysics that corresponds to the polytropic models. In this paper, a pseudospectral technique is proposed to solve the Lane–Emden type equations on a semi-infinite domain. The method is based on rational Legendre functions and Gauss–Radau integration. The method reduces solving the nonlinear ordinary differential equation to solve a system of nonlinear algebraic equations. The comparison of the results with the other numerical methods shows the efficiency and accuracy of this method.     

Influence of seed layers on the vertical growth of ZnO nanowires by low-temperature wet chemical bath deposition on ITO-coated glass substrate

Experimental Techniques - Aligned ZnO nanowires (NWs) were synthesized by low-temperature wet chemical bath deposition (CBD) technique on indium tin oxide (ITO)-coated glass substrates using c-axis...     

Accelerated double-step scale splitting iteration method for solving a class of complex symmetric linear systems

Numerical Algorithms - In this paper, we introduce and study an accelerated double-step scale splitting (ADSS) iteration method for solving complex linear systems. The convergence of the ADSS...     

Evaluation of catalytic activity of two newly prepared functionalized sulfonic acids ionic liquids in the synthesis of carbamatoalkyl naphthols under mild conditions

Russian Journal of General Chemistry - Catalytic activity of two newly prepared functionalized sulfonic acids ionic liquids, [MPyrrSO3H]Cl (IL1) and [MMorSO3H]Cl (IL2), is studied in one-pot...     

Three-Component,One-Pot Synthesis of 2,4,5-Trisubstituted Imidazoles Catalyzed by TiCl4-SiO2 Under Conventional Heating Conditions or Microwave Irradiation

In this study, silica-supported titanium tetrachloride was prepared and used as a novel catalyst for rapid and efficient synthesis of 2,4,5-trisubstituted imidazoles by a three-component, one-pot condensation of 1,2-diketones and aryl aldehydes in excellent yield under solvent-free conditions using conventional heating or microwave irradiation.