1D Generalised Burgers-Huxley: Proposed Solutions Revisited and Numerical Solution Using FTCS and NSFD Methods

In this paper, we obtain the numerical solution of a 1-D generalised Burgers-Huxley equation under specified initial and boundary conditions, considered in three different regimes. The methods are Forward Time Central Space (FTCS) and a non-standard finite difference scheme (NSFD). We showed the schemes satisfy the generic requirements of the finite difference method in solving a particular problem. There are two proposed solutions for this problem and we show that one of the proposed solutions contains a minor error. We present results using FTCS, NSFD, and exact solution as well as show how the profiles differ when the two proposed solutions are used. In this problem, the boundary conditions are obtained from the proposed solutions. Error analysis and convergence tests are performed.

Analysis and Assessment of the Effects of Corrosive Hydrogen Media on the Stress-Strain State of a Spherical Titanium Alloy Shell

We consider the creation of a mathematical model describing the effect of corrosive hydrogen environment on the stress state of a hollow spherical shell made of titanium alloy grade VT1-0, the load is evenly distributed throughout the shell. The solution of the problem in practice was carried out by two-step method of sequential perturbation of parameters using MatLab and Maple programs. To solve the system of solving differential equations the finite difference method was applied. The solution of the diffusion equation of the aggressive hydrogen medium has been considered and the obtained solution has been compared with the results of the classical theory which does not take into account the aggressive effect of the corrosive medium.

Fractal–Fractional Michaelis–Menten Enzymatic Reaction Model via Different Kernels

In this paper, three new models of fractal–fractional Michaelis–Menten enzymatic reaction (FFMMER) are studied. We present these models based on three different kernels, namely, power law, exponential decay, and Mittag-Leffler kernels. We construct three schema of successive approximations according to the theory of fractional calculus and with the help of Lagrange polynomials. The approximate solutions are compared with the resulting numerical solutions using the finite difference method (FDM). Because the approximate solutions in the classical case of the three models are very close to each other and almost matches, it is sufficient to compare one model, and the results were good. We investigate the effects of the fractal order and fractional order for all models. All calculations were performed using Mathematica software.

Mathematical model for calculating the temperature field of a direct-flow drying drum

In this paper, one parabolic-type boundary value problem is solved for determining the temperature field of the raw cotton and air components in drum dryers. In the proposed model, convective heat transfer is used according to Newton’s law, the terms describing the evaporation of moisture from the components of raw cotton (seeds, fiber) and the influence of air velocity are taken into account. The resulting system of Galerkin’s differential equations is solved by the finite-difference method in time. It is shown that the approximate solution is estimated according to Galerkin in Sobolev space.The numerical results of the considered problem are obtained by the Bubnov–Galerkin method. A comparative analysis is carried out with experimental data. It is shown that the proposed mathematical model and its numerical algorithm adequately describe the drying process of raw cotton.

Numerical analysis of shear interaction of an underground structure with soil

A soil layer behaviour under the shear interaction of an underground structure with soil is studied. Structural failure is considered under conditions of strained soil, and complete cohesion is assumed at the underground structure-soil contact boundary. The Finite Difference Method is used to numerically investigate the process of the structure-soil shear interaction under consideration. The main attention is paid to the adequacy of the conditions of soil-structure interaction, and to the strain state of the near-contact soil layer around the underground structure. The results are plotted and analysed. From the results obtained, the existence of a near-contact soil layer is shown; the use of the condition of complete cohesion is justified considering the structural failure of soil under conditions of complex interaction; the possibility to determine the thickness of the near-contact soil layer and of the layers with the respective degrees of structural failure is shown.

NUMERICAL RESULTS OF MIXED CONVECTION FLOW OVER A FLAT PLATE WITH THE IMPOSED HEAT AND ANGLE OF INCLINATION

In this paper, the numerical results of mixed convection flow over a flat plate with the imposed heat and different angles of inclination are established by applying the finite difference method of Crank-Nicolson. We further compare these numerical results with the case of non-mixed convection flow. The velocity and temperature profiles are decreased when the different values of the Prandtl number (Pr) are increased. Meanwhile, the velocity profiles are increased, when the different values of angle of inclination (alfa) and mixed convection parameter (lambda) are increased. The mixed convection flow case (lambda=1.5) is affected by the external force, so the velocity of convection flow is higher than the non-mixed case (lambda=0).

A new formulation of finite difference and finite volume methods for solving a space fractional convection–diffusion model with fewer error estimates

Convection and diffusion are two harmonious physical processes that transfer particles and physical quantities. This paper deals with a new aspect of solving the convection–diffusion equation in fractional order using the finite volume method and the finite difference method. In this context, we present an alternative way for estimating the space fractional derivative by utilizing the fractional Grünwald formula. The proposed methods are conditionally stable with second-order accuracy in space and first-order accuracy in time. Many comparisons are performed to display reliability and capability of the proposed methods. Furthermore, several results and conclusions are provided to indicate appropriateness of the finite volume method in solving the space fractional convection–diffusion equation compared with the finite difference method.

Kirchhoff-Love Plate Theory: First-Order Analysis, Second-Order Analysis, Plate Buckling Analysis, and Vibration Analysis Using the Finite Difference Method

This paper presents an approach to the Kirchhoff-Love plate theory (KLPT) using the finite difference method (FDM). The KLPT covers the case of small deflections, and shear deformations are not considered. The FDM is an approximate method for solving problems described with differential equations. The FDM does not involve solving differential equations; equations are formulated with values at selected points of the structure. Generally in the case of KLPT, the finite difference approximations are derived based on the fourth-order polynomial hypothesis (FOPH) and second-order polynomial hypothesis (SOPH) for the deflection surface. The FOPH is made for the fourth and third derivative of the deflection surface while the SOPH is made for its second and first derivative; this leads to a 13-point stencil for the governing equation. In addition, the boundary conditions and not the governing equations are applied at the plate edges. In this paper, the FOPH was made for all of the derivatives of the deflection surface; this led to a 25-point stencil for the governing equation. Furthermore, additional nodes were introduced at plate edges and at positions of discontinuity (continuous supports/hinges, incorporated beams, stiffeners, brutal change of stiffness, etc.), the number of additional nodes corresponding to the number of boundary conditions at the node of interest. The introduction of additional nodes allowed us to apply the governing equations at the plate edges and to satisfy the boundary and continuity conditions. First-order analysis, second-order analysis, buckling analysis, and vibration analysis of plates were conducted with this model. Moreover, plates of varying thickness and plates with stiffeners were analyzed. Finally, a direct time integration method (DTIM) was presented. The FDM-based DTIM enabled the analysis of forced vibration of structures, with damping taken into account. In first-order, second-order, buckling, and vibration analyses of rectangular plates, the results obtained in this paper were in good agreement with those of well-established methods, and the accuracy was increased through a grid refinement.

Development Length of Fluids Modelled by the gPTT Constitutive Differential Equation

In this work, we present a numerical study on the development length (the length from the channel inlet required for the velocity to reach 99% of its fully-developed value) of a pressure-driven viscoelastic fluid flow (between parallel plates) modelled by the generalised Phan–Thien and Tanner (gPTT) constitutive equation. The governing equations are solved using the finite-difference method, and, a thorough analysis on the effect of the model parameters α and β is presented. The numerical results showed that in the creeping flow limit (Re=0), the development length for the velocity exhibits a non-monotonic behaviour. The development length increases with Wi. For low values of Wi, the highest value of the development length is obtained for α=β=0.5; for high values of Wi, the highest value of the development length is obtained for α=β=1.5. This work also considers the influence of the elasticity number.