Numerical integration method for two-parameter singularly perturbed time delay parabolic problem

We design a suitable fitted operator finite difference method (FOFDM) to solve a model of an unsteady transient magnetohydrodynamic (MHD) free convective and mass transfer flow with thermophoresis past an inclined permeable plate in the presence of chemical reaction, thermal radiation, and temperature dependent viscosity. The governing nonlinear partial differential equations in the current model are transformed by a suitable similarity transformation into a system of ordinary differential equations which are then solved numerically using the FOFDM. The superiority of the proposed FOFDM over the standard finite difference method is proved theoretically and numerically. Furthermore, we illustrate the effect of various parameters on the temperature, concentration and velocity profiles using the FOFDM. It is observed that an increase in the viscosity parameter leads to a decrease in the temperature and an increase in the velocity. Results also show that an increase in the thermophoretic parameter leads to a decrease in the concentration. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 106–120, 2016

This paper is concerned with the numerical solution of the singularly perturbed differential-difference equations with small shifts called delay and advanced parameters. A fourth order finite difference method with a fitting factor is proposed for the solution of the singularly perturbed differential-difference equations with mixed shifts. The delay and advanced shifts are managed by Taylor series and an asymptotically equivalent singularly perturbed two-point boundary value problem is obtained. A fitting factor is introduced in the fourth order finite difference scheme for the problem which takes care of the small values of the perturbation parameter. This fitting factor is obtained from the asymptotic solution of singular perturbations. Thomas algorithm is used to solve the discrete system of the difference scheme. Convergence of the proposed method is analyzed. Maximum absolute errors in comparison with the several numerical experiments aretabulated to illustrate the proposed method.

In this paper, we extend a system of coupled first order non-linear system of delay differential equations (DDEs) arising in modeling of stoichiometry of tumour dynamics, to a system of diffusion-reaction system of partial delay differential equations (PDDEs). Since tumor cells are further modified by blood supply through the vascularization process, we determine the local uniform steady states of the homogeneous tumour growth model with respect to the vascularization process. We show that the steady states are globally stable, determine the existence of Hopf bifurcation of the homogeneous tumour growth model with respect to the vascularization process. We derive, analyse and implement a fitted operator finite difference method (FOFDM) to solve the extended model. This FOFDM is analyzed for convergence and we observe seen that it has second-order accuracy. Some numerical results confirming theoretical observations are also presented. These results are comparable with those obtained in the literature.

In this paper, accelerated fitted finite difference method for solving singularly perturbed delay differential equation with non-local boundary condition is considered. To treat the non-local boundary condition, Simpson’s rule is applied. The stability and parameter uniform convergence for the proposed method are proved. To validate the applicability of the scheme, two model problems are considered for numerical experimentation and solved for different values of the perturbation parameter ε and mesh size h. The numerical results are tabulated in terms of maximum absolute errors and rate of convergence, and it is observed that the present method is more accurate and ε-uniformly convergent for h ≥ ε where the classical numerical methods fails to give good result, and it also improves the results of the methods existing in the literature.

This article addresses the abilities and limitations of the Lattice Boltzmann (LB) method in solving advection-dominated mass transport problems. Several schemes of the LB method, including D2Q4, D2Q5, and D2Q9, were assessed in the simulation of two-dimensional advection-dispersion equations. The concept of Single Relaxation Time (SRT) and Multiple Relaxation Time (MRT) in addition to linear and quadratic Equilibrium Distribution Functions (EDF) were taken into account. The results of LB models were compared to the well-known Finite Difference (FD) solutions, including Explicit Finite Difference (EFD) and Crank-Nicolson (CN) methods. All LB models are more accurate than the aforementioned FD schemes. The results also indicate the high potency of D2Q5 SRT and D2Q9 SRT in describing advection-controlled mass transfer problems. The numerical artificial oscillations are observed when the Grid Peclet Number (GPN) is greater than 10, 25, 20, 25, and 10 regarding D2Q4 SRT, D2Q5 SRT, D2Q5 MRT, D2Q9 SRT and D2Q9 MRT, respectively, while the corresponding GPN values obtained for the EFD and CN methods were 2 and 5, respectively. Finally, a coupled system of groundwater and solute transport equations were solved satisfactorily using several LB models. Considering computational time, all LB models are much faster than CN method.

In this study, a singularly perturbed linear impulsive system with singularly perturbed impulses is considered. Many books discuss different types of singular perturbation problems. In the present work, an impulse system is considered in which a small parameter is introduced into the impulse equation. This is the main novelty of our study, since other works [25] have only considered a small parameter in the differential equation. A necessary condition is also established to prevent the impulse function from bloating as the parameter approaches zero. As a result, the notion of singularity for discontinuous dynamics is greatly extended. An asymptotic expansion of the solution of a singularly perturbed initial problem with an arbitrary degree of accuracy for a small parameter is constructed. A theorem for estimating the residual term of the asymptotic expansion is formulated, which estimates the difference between the exact solution and its approximation. The results extend those of [32], which formulates an analogue of Tikhonov’s limit transition theorem. The theoretical results are confirmed by a modelling example.

The theory of singular perturbations and time scales has played an important role in aircraft performance analysis and in the development of feedback guidance laws for aerospace vehicles. This paper presents a survey of the applications of the theory of singular perturbations and time scales in guidance, navigation, and control of air and space transportation systems. In particular, emphasis will be placed on problem formulation and solution approaches that are useful in applying the theory for various types of problems arising in aerospace systems. Fhrther, the survey e m braces the related fields such as fluid dynamics, space structures and space robotics.

The Crank–Nicolson method and a further method, called the Richtmyer–Morton method, in which $u_t (t)$ is approximated by ${{[3u(t) - 4u(t - \Delta t) + u(t - 2\Delta t)]} / {2\Delta t}}$, are applied to one-dimensional parabolic initial-boundary value problems for $u_t = L(x,t;u)$ with strongly elliptic operator L. It is shown for time-independent linear L that these methods are A-convergent and accurate to $O(\Delta x^2 + \Delta t^2 )$ in the max norm. In the case of time-dependent weakly nonlinear L, relative to the discrete $L_2 $-norm, the same result is derived for the Crank–Nicolson method whereas the error of the Richtmyer–Morton method on the line $t = T$ is revealed to be accurate to \[ T^{{1 / 2}} O\left( {\frac{1} {{\Delta t^{{1 / 2}} }}\left( {\Delta x^2 + \Delta t^2 } \right)} \right). \]

In this article, a uniformly convergent numerical scheme is developed to solve a singularly perturbed convection‐diffusion equation with a small delay having a boundary layer along the left side. A priori bounds of continuous solution and its derivatives are discussed. To solve the problem, the Crank–Nicolson scheme in the time direction and the exponentially fitted finite difference scheme in the space direction are used. The stability of the method is analyzed. It is proved that the developed scheme converges uniformly with first order in space and second order in time. To validate the applicability of the theoretical finding of the developed scheme, numerical experiments are carried out by considering two test examples.

A parameter-robust finite difference method for singularly perturbed delay parabolic partial differential equations

The fundamental study of the phenomenon of natural convective heat and mass transfer near a vertical plate in the presence of porous medium is investigated. The fluids considered in the discussion are air and water. The unsteady, coupled, non-linear partial differential equations are solved numerically by Crank Nicolson scheme. The influence of various parameters like velocity, temperature and concentration are analyzed. The skin friction, the Nusselt number, the Sherwood number, average skin friction, average Nusselt number and average Sherwood number are also presented graphically for both air and water. It has been observed that velocity components increase with permeability parameter where as temperature and concentration decreases with it. An analysis is performed to check the stability and the convergence of the finite difference scheme which converges faster and unconditionally stable.

The error generated by the classical upwind finite difference method on a uniform mesh, when applied to a class of singularly perturbed modelordinary different alequations with a singularly perturbed Neumann boundary condition, tends to infinity as the singular perturbation parameter tends to zero. Note that the exact solution is uniformly bounded with respect to the perturbat on parameter. For the same classcal finite difference operator on an appropriate piecewise -uniform mesh, it is shown that the numerical solutions converge, uniformly with respect to the perturbation parameter, to the exact solution of any problem from this class.KeywordsNeumann Boundary ConditionPerturbation ParameterUniform MeshDiscrete SolutionMesh RegionThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

SIMPLE SUMMARY: The use of flaxseed in complete pet foods as a source of omega-3 fatty acids, and in particular alpha-linolenic acid, has been studied extensively in dogs. However, to date, no scientific published literature exists on the impacts of feeding whole ground flaxseed as an ingredient in cat food on nutrient digestibility, fatty acid bioavailability, and overall health in cats. In addition, dietary flaxseed inclusion has been challenged as providing little nutritional value for cats by not changing serum fatty acid levels and lowering nutrient digestibility. Therefore, the objective of this study was to directly evaluate the effects of adding ground flaxseed to dry cat food on fatty acid bioavailability, stool consistency, nutrient digestibility, and overall health in adult cats. These results clearly exhibit that a flaxseed-supplemented diet resulted in no changes in stool quality or nutrient digestibility, and elicited an increase in serum alpha-linolenic acid concentrations in healthy cats. Overall, these results clearly demonstrate that ground flaxseed can be used to modulate serum fatty acid concentrations and serve as a bioavailable source of alpha-linolenic acid in commercial cat food. ABSTRACT: We evaluated effects of dietary ground flaxseed on fecal and serum alpha-linolenic acid (ALA) concentrations, nutrient digestibility, and stool quality in female and male adult cats (n = 20 (8 males, 12 females); 3.95 ± 1.49 years of age (mean ± SD); 3.88 ± 0.82 kg BW). We hypothesized that adding ground flaxseed would increase serum ALA compared with feeding no flax, without changing nutrient digestibility. Cats were fed as-is 2.6% added-flaxseed (flax, n = 10) or no-flax (control, n = 10) diets (2.66 vs. 0.78% ALA of total fatty acids; crude protein 35%, fat 20%, fiber 3% as-fed) twice daily to maintain body weight for 28 days. Fecal collections were conducted on days 23–27 for total-tract nutrient digestibility, stool quality (scale 1–5; 1 = watery diarrhea, 5 = hard, dry, crumbly) and long-chain fatty acid (LCFA) analyses. Blood was collected on days 0, 14, and 28 for serum LCFA and chemistry analysis. Digestibility and fecal data were analyzed by ANOVA (SAS v9.4, Cary, NC, USA) and a repeated measures ANOVA for serum ALA. Flax-fed cats, compared with control-fed, had greater (p < 0.05) serum ALA after 14 days (4.00 vs. 0.71 µg/mL) and 28 days (7.83 and 3.67 µg/mL). No differences were observed in stool quality, and dry matter, protein, fat, and ALA digestibility. However, metabolizable energy was greater in the flax vs. control diet (4.18 vs. 3.91 kcal/g; p < 0.05). Overall, these data demonstrate that ground flaxseed added to cat diets increases serum ALA within 14 days, with no detriments to nutrient digestibility. We conclude that flaxseed can be used as a bioavailable source of ALA in cat diets.

In this paper the H/sub /spl infin//-optimal control and risk-sensitive control of linear singularly perturbed, discrete-time systems are considered. It is shown that the Riccati equation associated with the solution of the H/sub /spl infin//-optimal control problem, can be approximated by an outer series solution, and a boundary-layer correction series solution. As the singular perturbation parameter /spl epsiv/ tends to zero, it is shown that the full-order Riccati equation can be approximated by the zeroth-order outer-series solution. It is also recognized that the outer-series solution and the boundary-layer solution, are equivalent to the resulting asymptotic series expansion in powers of /spl epsiv/, of the Riccati equation associated with a certain risk-sensitive control problem. When a two-time-scale decomposition is considered, it is shown that an approximate composite control can be formed, as the sum of the slow and fast subsystems. Unlike the asymptotic expansion approach, it is shown that the original system can be approximated by the maximum of the performance associated with the slow and fast subsystems. Finally, it is shown that the slow and fast subsystems, are equivalent to certain slow and fast subsystems, respectively, resulting from applying a two-time-scale decomposition to a certain risk-sensitive control problem. >

Some Topics on Variational Inequalities and Applications