Scheme For Ordinary Differential Equations Research Articles

Strong stochastic Runge-Kutta–Munthe-Kaas methods for nonlinear Itô SDEs on manifolds

In this paper we present numerical methods for the approximation of nonlinear Itô stochastic differential equations on manifolds. For this purpose, we extend Runge-Kutta–Munthe-Kaas (RKMK) schemes for ordinary differential equations on manifolds to the stochastic case and analyse the strong convergence of these schemes. Since these schemes are based on the application of a stochastic Runge-Kutta (SRK) scheme in a corresponding Lie algebra, we address the question under which circumstances the stochastic RKMK method has the same strong order of convergence as the applied SRK scheme. To illustrate our answer to this question and the effectiveness of our schemes, we show some numerical results of applying these methods to a problem with an autonomous underwater vehicle.

Applications of triadic hybridized-cross nanomaterials suspended in engine oil using quadratic and linear convection with magnetic dipole

This work comprises the contribution of buoyancy forces, linear and quadratic convection on cross Nano fluid under the effects of dipole effects with chemical reaction. The flow of a cross model (CM) with mass and thermal transport has been modeled in Cartesian coordinates for the two-dimensional fluid flow problem. Temperature gradient effects have been incorporated in mass transport and concentration gradient impacts in thermal transport phenomenon with chemical reaction. The governing laws are derived under boundary layer principle (BLP) in the conformation of coupled nonlinear partial differential equations. Afterwards, coupled partial differential equations are converted into a scheme of ordinary differential equations (ODEs) by incorporating the similarity transformation. The nonlinear modeled equations are solved numerically with the aid of finite element scheme as well as analysis of grid independent is reported based on the reliability of the utilized approach. Several graphs and tables have been prepared to observe the bearing of multiple emerging parameters regarding velocity, temperature as well as concentration fields. Comparison has been made to confirm the efficiency of a finite element scheme. It is perceived that higher values of a Schmidt number control the fluid concentration. Moreover, velocity is diminished against higher values of Weissenberg parameter.

High-order linearly implicit schemes conserving quadratic invariants

In this paper, we propose linearly implicit and arbitrary high-order conservative numerical schemes for ordinary differential equations with a quadratic invariant. Many differential equations have invariants, and numerical schemes for preserving them have been extensively studied. Since linear invariants can be easily kept after discretisation, quadratic invariants are essentially the simplest ones. Quadratic invariants are important objects that appear not only in many physical examples but also in the computationally efficient conservative schemes for general invariants such as scalar auxiliary variable approach, which have been studied in recent years. It is known that quadratic invariants can be maintained relatively easily compared with general invariants, and can be preserved by canonical Runge–Kutta methods. However, there is no unified method for constructing linearly implicit and high order conservative schemes. In this paper, we construct such schemes based on canonical Runge–Kutta methods and prove some properties involving accuracy.

Buoyancy Effect on the Unsteady Diffusive Convective Flow of a Carreau Fluid Passed over a Coated Disk with Energy Loss

The unsteady flow of a Carreau fluid over a coated disk under the simultaneous effects of a thermal and concentration field with buoyancy forces is reported. The time-dependent diffusive stream of a Carreau fluid over a conducting coated disk is carried out with energy loss. The time-dependent partial differential equations are first converted into a scheme of ordinary differential equations by the appropriate transformations and are then solved by shooting method. Significant results for speed, hotness and concentration profiles are revealed and deliberated by the graphical outcomes. The numerical values of skin friction suggest that the viscoelastic parameter of the Carreau fluid causes a reduction in the skin friction coefficient due to the coated surface, but the Nusselt and Sherwood numbers increase with the rise of the viscoelastic parameter of the Carreau fluid because of the coated surface. The present model is useful in the field of mechanical engineering to design a tesla turbine for the flow of viscous fluid.

On the Stability and Numerical Scheme of Fractional Differential Equations with Application to Biology

The fractional differential equations involving different types of fractional derivatives are currently used in many fields of science and engineering. Therefore, the first purpose of this study is to investigate the qualitative properties including the stability, asymptotic stability, as well as Mittag–Leffler stability of solutions of fractional differential equations with the new generalized Hattaf fractional derivative, which encompasses the popular forms of fractional derivatives with non-singular kernels. These qualitative properties are obtained by constructing a suitable Lyapunov function. Furthermore, the second aim is to develop a new numerical method in order to approximate the solutions of such types of equations. The developed method recovers the classical Euler numerical scheme for ordinary differential equations. Finally, the obtained analytical and numerical results are applied to a biological nonlinear system arising from epidemiology.

Invariant variational schemes for ordinary differential equations

AbstractWe propose a novel algorithmic method for constructing invariant variational schemes of systems of ordinary differential equations that are the Euler–Lagrange equations of a variational principle. The method is based on the invariantization of standard, noninvariant discrete Lagrangian functionals using equivariant moving frames. The invariant variational schemes are given by the Euler–Lagrange equations of the corresponding invariantized discrete Lagrangian functionals. We showcase this general method by constructing invariant variational schemes of ordinary differential equations that preserve variational and divergence symmetries of the associated continuous Lagrangians. Noether's theorem automatically implies that the resulting schemes are exactly conservative. Numerical simulations are carried out and show that these invariant variational schemes outperform standard numerical discretizations.

An adaptive BDF2 implicit time-stepping method for the phase field crystal model

Abstract An adaptive BDF2 implicit time-stepping method is analyzed for the phase field crystal model. The suggested method is proved to preserve a modified energy dissipation law at the discrete levels when the time-step ratios satisfy $r_k:=\tau _k/\tau _{k-1}<3.561$, which is the zero-stability restriction of the variable-step BDF2 scheme for ordinary differential equations. With the help of discrete orthogonal convolution kernels and corresponding convolution inequalities, an optimal $L^2$ norm error estimate is established under the weak step-ratio restriction $0<r_k<3.561$ to ensure energy stability. As far as we know, this is the first time that such an error estimate is theoretically proved for a nonlinear parabolic equation. Based on tests on random temporal meshes an effective adaptive time-stepping strategy is suggested to efficiently capture the multi-scale behavior and accelerate the numerical simulations.

Stagnation point flow of nanofluid containing micro-organisms

The present study focus on the magnetohydrodynamic nanofluid containing micro-organisms flow in the subsistence of thermophoresis and Brownian action event. Furthermore, Dufour and Soret effects have been taken into detail through the heat and mass transfer analysis. The coordination of partial differential equations renewed into a scheme of ordinary differential equations using the similarity renovation and also numerically solved via bvp4c MATLAB package. The physical parameters are discussed through different graphs and tabular form. Moreover, it is exposed that the raising concentration and temperature with greater utility of Dufour and Soret effect. For large values in fluid concentration for greater change in activation parameter. The larger values of Peclet number and concentration difference the density quantity of motile micro-organisms increases.

On the development of symmetry-preserving finite element schemes for ordinary differential equations

In this paper we introduce a procedure, based on the method of equivariant moving frames, for formulating continuous Galerkin finite element schemes that preserve the Lie point symmetries of initial value problems for ordinary differential equations. Our methodology applies to projectable and non-projectable symmetry group actions, to ordinary differential equations of arbitrary order, and finite element approximations of arbitrary polynomial degree. Several examples are included to illustrate various features of the symmetry-preserving process. We summarise extensive numerical experiments showing that symmetry-preserving finite element schemes may provide better long term accuracy than their non-invariant counterparts and can be implemented on larger elements.

Melting heat transfer with radiative effects and homogeneous–heterogeneous reaction in thermally stratified stagnation flow embedded in porous medium

The present article deals with thermally stratified stagnation-point flow saturated in porous medium on surface of variable thickness along with more convincing and reliable surface condition termed as melting heat transfer. Homogeneous–heterogeneous reaction and radiative effects have been further taken into account to reconnoiterproperties of heat transfer. Melting heat transfer and phenomenon of homogeneous–heterogeneous reaction have engrossed widespread utilization in purification of metals, welding process, electroslag melting, biochemical systems, catalysis and several industrial developments. Suitable transformations are utilized to attain a scheme of ordinary differential equations possessing exceedingly nonlinear nature. Homotopic process is employed to develop convergent solutions of the resulting problem. Discussion regarding velocity, thermal field and concentration distribution for several involved parameters is pivotal part. Graphical behaviors of skin friction coefficient and Nusselt number are also portrayed. Concentration of the reactants is found to depreciate as a result of strength of both heterogeneous and homogeneous reaction parameters. With existence of melting phenomenon, declining attitude of fluid temperature is observed for higher radiation parameter.

Higher-order additive Runge–Kutta schemes for ordinary differential equations

Two new implicit–explicit, additive Runge–Kutta (ARK2) methods are given with fourth- and fifth-order formal accuracies, respectively. Both combine explicit Runge–Kutta (ERK) methods with explicit, singly-diagonally implicit Runge–Kutta (ESDIRK) methods and include an embedded method for error control. The two methods have ESDIRKs which are internally L-stable on stages three and higher, have only modestly negative eigenvalues to the stage and step algebraic-stability matrices and have stage-order two. To improve computational efficiency, the fourth-order method has a diagonal coefficient of 0.1235. This is done to offset much of the extra computational cost of an extra stage by facilitating iterative convergence at each stage. Linear stability domains for both ERK methods have been made quite large and the dominant coupling stability term between the stability of the ESDIRK and ERK for very stiff modes has been removed. Though the fourth-order method is one of the best all-around fourth-order IMEX ARK2 of which we are aware, the fifth-order method is likely best suited to mildly stiff problems with tight error tolerances. Methods are tested using the Van der Pol and Kaps' singular-perturbation problems. Results suggest that these new methods represent an improvement over existing methods of the same class.

Quantification of kinetic rate constants for transcytosis of polymeric nanoparticle through blood-brain barrier

BackgroundPolymeric nanoparticles (PNP) have received significant amount of interests for targeted drug delivery across the blood-brain barrier (BBB). Experimental studies have revealed that PNP can transport drug molecules from microvascular blood vessels to brain parenchyma in an efficient and non-invasive way. Despite that, very little attention has been paid to theoretically quantify the transport of such nanoparticles across BBB. MethodsIn this study, for the first time, we developed a mathematical model for PNP transport through BBB endothelial cells. The mathematical model is developed based on mass-action laws, where kinetic rate parameters are determined by an artificial neural network (ANN) model using experimental data from in-vitro BBB experiments. ResultsThe presented ANN model provides a much simpler way to solve the parameter estimation problem by avoiding integration scheme for ordinary differential equations associated with the mass-action laws. Furthermore, this method can efficiently deal with both small and large data set and can approximate highly nonlinear functions. Our results show that the mass-action model, constructed with ANN based rate parameters, can successfully predict the characteristics of the polymeric nanoparticle transport across the BBB. ConclusionsOur model results indicate that exocytosis of nanoparticles is seven fold slower to endocytosis suggesting that future studies should focus on enhancing the exocytosis process. General significanceThis mathematical study will assist in designing new drug carriers to overcome the drug delivery problems in brain. Furthermore, we anticipate that this model will form the basis of future comprehensive models for drug transport across BBB.

MHD micropolar fluid flow over a stretching permeable sheet in the presence of thermal radiation and thermal slip flow: a numerical study

In this manuscript, a mathematical explanation is an attempt at meant for two-dimensional, micropolar fluid flow over a permeable stretching sheet with viscous dissipation in the occurrence of thermal radiation and temperature dependent slip flow. With the similarity transformations, the governing equations have been changed into a scheme of ordinary differential equations. These differential equations are extremely nonlinear which cannot be solved analytically. Thus, bvp4c MATLAB solvers have been used for solving it. Numerical consequences are obtained for the skin-friction coefficient, the couple wall stress and the local Nusselt number with the same as velocity, microrotation and temperature profiles for various values of the governing parameters, namely, material factor, magnetic factor, thermal slip factor, radiation factor, Prandtl number and Eckert number. It is found that magnetic field reduces the fluid velocity and angular velocity, but magnetic field enhances the fluid temperature. Furthermore, fluid temperature increases with increases in thermal slip parameter.

Error Inhibiting Block One-step Schemes for Ordinary Differential Equations

The commonly used one step methods and linear multi-step methods all have a global error that is of the same order as the local truncation error (as defined in [1, 6, 8, 13, 15]). In fact, this is true of the entire class of general linear methods. In practice, this means that the order of the method is typically defined solely by order conditions which are derived by studying the local truncation error. In this work we investigate the interplay between the local truncation error and the global error, and develop a methodology which defines the construction of explicit error inhibiting block one-step methods (alternatively written as explicit general linear methods [2]). These error inhibiting schemes are constructed so that the accumulation of the local truncation error over time is controlled, which results in a global error that is one order higher than the local truncation error. In this work, we delineate how to carefully choose the coefficient matrices so that the growth of the local truncation error is inhibited. We then use this theoretical understanding to construct several methods that have higher order global error than local truncation error, and demonstrate their enhanced order of accuracy on test cases. These methods demonstrate that the error inhibiting concept is realizable. Future work will further develop new error inhibiting methods and will analyze the computational efficiency and linear stability properties of these methods.

Consistency and asymptotic normality of stochastic Euler schemes for ordinary differential equations

General stochastic Euler schemes for ordinary differential equations are studied. We give proofs on the consistency, the rate of convergence and the asymptotic normality of these procedures.

Stabilization by Artificial Wiener Processes

In this technical note, we restage the positive effects of Gaussian white noises for continuous-time driftless input-affine nonholonomic systems. Compared with previous studies, we found that the positive effects are hidden in the shadows of nonuniqueness of stochastic integrals over multi-dimensional Wiener processes. Furthermore, we found that the positive effects are artificially restaged by numerical calculations with use of Euler scheme for ordinary differential equations. Our results demonstrate that the positive effects act as “hidden independent control inputs” for nonholonomic systems. This implies that our results provide a constructive procedure for designing control inputs such that the origins become $p$ -th moment exponentially or finite-time stable.

Erratum to: A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. In multibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group $$\textit{SO}\,(3) \times {\mathbb {R}}^{3}$$ , although rigid body motions form the semidirect product Lie group $$\textit{SE}\,(3) $$ . It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of $$\textit{SE}\,(3) $$ have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that $$\textit{SE}\,(3) $$ is the appropriate c-space for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

A note on the motion representation and configuration update in time stepping schemes for the constrained rigid body

The dynamics of a holonomically constrained rigid body can be modeled by Newton-Euler equations subjected to geometric constraints. This is frequently formulated as a differential-algebraic equation (DAE) system of index 1. In multibody system (MBS) dynamics it is common (1) to numerically solve this system by means of integration schemes for ordinary differential equations, and (2) to treat the rigid body motion on the direct product Lie group $$\textit{SO}\,(3) \times {\mathbb {R}}^{3}$$ , although rigid body motions form the semidirect product Lie group $$\textit{SE}\,(3) $$ . It is has been observed that the constraint satisfaction depends on which Lie group is used as configuration space (c-space). In this paper the problem is considered from a geometric perspective. It is shown that the constraints are exactly satisfied by a numerical integration scheme if they define a subgroup of the c-space. The subgroups of $$\textit{SE}\,(3) $$ have a significance for modeling mechanical systems, including lower kinematic (Reuleaux) pairs and are implicitly used in MBS modeling. It is concluded that $$\textit{SE}\,(3) $$ is the appropriate c-space for numerical DAE modeling of a constrained rigid body. This result does not immediately apply to MBS, however.

A posteriori error analysis for finite element methods with projection operators as applied to explicit time integration techniques

We derive a posteriori error estimates for two classes of explicit finite difference schemes for ordinary differential equations. To facilitate the analysis, we derive a systematic reformulation of the finite difference schemes as finite element methods. The a posteriori error estimates quantify various sources of discretization errors, including effects arising from explicit discretization. This provides a way to judge the relative sizes of the contributions, which in turn can be used to guide the choice of various discretization parameters in order to achieve accuracy in an efficient way. We demonstrate the accuracy of the estimate and the behavior of various error contributions in a set of numerical examples.

A note on the stability of the upwind scheme for ordinary differential equations

In this paper the stability of the upwind scheme for second order, linear ordinary differential equations with general boundary conditions of the third kind is proved. The proof is carried out via compactness arguments and is rather short. Moreover, no information on the behaviour of the solution of the differential equation itself is needed. This approach seems not to be present in the existing literature.