In this paper, recommended spiral passive micromixer was designed and simulated. spiral design has the potential to create and strengthen the centrifugal force and the secondary flow. A series of simulations were carried out to evaluate the effects of channel width, channel depth, the gap between loops, and flowrate on the micromixer performance. These features impact the contact area of the two fluids and ultimately lead to an increment in the quality of the mixture. In this study, for the flow rate of 25 μl/min and molecular diffusion coefficient of 1×10-10 m2/s, mixing efficiency of more than 90% is achieved after 30 (approximately one-third of the total channel length). Finally, the optimized design fabricated using proposed 3D printing method.

This article is concerned with using a finite difference method, namely the θ-methods, to solve the diffusion-convection equation with piecewise constant arguments. The stability of this scheme is also obtained. Since there are not many published results on the Numerical solution of this sort of differential equation and because of the importance of the above equation in the physics and engineering sciences, we have decided to study and present a stable Numerical solution for the above mentioned problem. At the end of article some experiments are done to demonstrate the stability of the scheme. We also draw the figures for the Numerical and analytical solutions which confirm our results. The Numerical solutions have also been compared with analytical solutions.

This paper is concerned with Numerical approach for solving space fractional diffusion equation using shifted Gegenbauer polynomials, where the fractional derivatives are expressed in Caputo sense. The properties of Gegenbauer polynomials are exploited to reduce space fractional diffusion equation to a system of ordinary differential equations, that are then solved using finite difference method. Some selected Numerical simulations of space fractional diffusion equations are presented and the results are compared with the exact solution, also with the results obtained via other methods in the literature. The comparison reveals that the proposed method is reliable, effective and accurate. All the computations were carried out using Matlab package.

boundary value problems. In general, classical Numerical methods fail to produce good approximations for the singular boundary value problems. In this paper, Chebyshev finite difference (ChFD) method and DTM-Pad’e method, which is a combination of differential transform method (DTM) and Pad´e approximant, are applied for solving singular boundary value problems arising in the reaction cum diffusion process in a spherical biocatalyst. ChFD method can be regarded as a non-uniform finite difference scheme and DTM is a Numerical method based on the Taylor series expansion, which constructs an analytical solution in the form of a polynomial. The main advantage of DTM is that it can be applied directly to nonlinear ordinary without requiring linearization, discretization or perturbation. Therefore, it is not affected by errors associated to discretization. The results obtained, are in good agreement with those obtained Numerically or by optimal homotopy analysis method.

In this paper, a high-order Numerical method is designed and implemented to solve a boundary value problem governed by the variable-order fractional diffusion equation. This equation contains a variable-order fractional time-derivative and a second-order spatial-derivative. To develop this novel method, a compact finite difference formula and a weighted shifted Grunwald-Letnikov operator are used for spatial and temporal discretization, respectively. It is shown that this method is of fourth- and second-order of convergence accuracy in spatial and time directions, respectively. Also, the solvability, stability and convergence of the peresent method are investigated. To verify the efficiency and high accuracy of this method, some Numerical examples and comparative results are presented.

In this paper, we derive a novel Numerical method to find out the Numerical solution of fractional partial differential equations (PDEs) involving Caputo-Fabrizio (C-F) fractional derivatives. We first find out the approximation formula of C-F derivative of function tk. We approximate the C-F derivative in time with the help of the Legendre spectral method and approximation formula of tk. The unknown function and their derivatives in spatial direction are approximated with the quasi wavelet-based Numerical method. We apply this newly derived method to solve the nonlinear distributed order reaction-diffusion in which time-fractional derivative is of C-F type. The accuracy and validity of the proposed method is exhibited by giving a solution to some Numerical examples. The obtained Numerical results are compared with the analytical results and conclude that our proposed Numerical method achieves accurate results. On the other hand, the method is easy to apply on higher-order fractional partial differential equations and variable-order fractional partial differential equations.

This paper deals with the Numerical treatment of two-parametric singularly perturbed parabolic convection-diffusion problems. The scheme is developed through the Crank-Nicholson discretization method in the temporal dimension followed by fitting the B-spline collocation method in the spatial direction. The effect of the perturbation parameters on the solution profile of the problem is controlled by fitting a parameter. As a result, it has been observed that the method is a parameter-uniform convergent and its order of convergence is two. This is shown by the boundedness of the solution, its derivatives, and error estimation. The effectiveness of the proposed method is demonstrated by model Numerical examples, and more accurate solutions are obtained as compared to previous findings available in the literature.

In this essay, we study the Numerical solution of Convection-diffusion equation with a memory term subject to initial boundary value conditions. Finite difference method in combination with product trapezoidal integration rule is used to discretize the equation in time and sinc collocation method is employed in space. The accuracy and error analysis of the method are discussed. Numerical examples and illustrations are presented to prove the validity of the suggested method.