Original Partial Differential Equation Research Articles

A PDE multiscale model of hepatitis C virus infection can be transformed to a system of ODEs

Direct-acting antivirals (DAAs) treat hepatitis C virus (HCV) by targeting its intracellular viral replication. DAAs are effective and deliver high clinical performance against HCV infection, but optimization of the DAA treatment regimen is ongoing. Different classes of DAAs are currently under development, and HCV treatments that combine two or three DAAs with different action mechanisms are being improved. To accurately quantify the antiviral effect of these DAA treatments and optimize multi-drug combinations, we must describe the intracellular viral replication processes corresponding to the action mechanisms by multiscale mathematical models. Previous multiscale models of HCV treatment have been formulated by partial differential equations (PDEs). However, estimating the parameters from clinical datasets requires comprehensive numerical PDE computations that are time consuming and often converge poorly. Here, we propose a user-friendly approach that transforms a standard PDE multiscale model of HCV infection (Guedj J et al., Proc. Natl. Acad. Sci. USA 2013; 110(10):3991–6) to mathematically identical ordinary differential equations (ODEs) without any assumptions. We also confirm consistency between the numerical solutions of our transformed ODE model and the original PDE model. This relationship between a detailed structured model and a simple model is called ``model aggregation problem'' and a fundamental important in theoretical biology. In particular, as the parameters of ODEs can be estimated by already established methods, our transformed ODE model and its modified version avoid the time-consuming computations and are broadly available for further data analysis.

On the analytical solutions of conformable time-fractional extended Zakharov–Kuznetsov equation through ( $$G'/G^{2}$$ G ′ / G 2 )-expansion method and the modified Kudryashov method

The aim of this article is to obtain and study the solutions of conformable time fractional (1+2)-dimensional extended Zakharov–Kuznetsov equation (2D-FZKE) which is modeled to investigate the waves in magnetized plasma. Wave transformation in fractional form is applied to convert the original fractional order nonlinear partial differential equation into another nonlinear ordinary differential equation. The strategy here consists of using ( $$G'/G^{2}$$ )-expansion method and the modified Kudryashov method to obtain a variety of exact solutions. Both schemes work well and reveal distinct exact solutions. These solutions are of significant importance in plasma physics where the 2D-FZKE is modeled for some special physical phenomenon.

Boundary control of an Euler–Bernoulli beam with input and output restrictions

In this study, boundary control is considered for an Euler–Bernoulli beam subject to bounded input, bounded output, and external disturbances. Through utilizing the backstepping technology, a boundary control scheme is designed based on the original partial differential equations to regulate the vibration of the beam. An auxiliary system based on a smooth hyperbolic function is designed to handle the impact of the restricted input. And a barrier Lyapunov function is adopted to eliminate the impact of output restriction. It is proved that the input and output restrictions are circumvented simultaneously. Simulations are demonstrated for illustration.

Exact and numerical solutions of time-fractional advection–diffusion equation with a nonlinear source term by means of the Lie symmetries

In this paper, the authors analyze a time-fractional advection–diffusion equation, involving the Riemann–Liouville derivative, with a nonlinear source term. They determine the Lie symmetries and reduce the original fractional partial differential equation to a fractional ordinary differential equation. The authors solve the reduced fractional equation adopting the Caputo’s definition of derivatives of non-integer order in such a way the initial conditions have a physical meaning. The reduced fractional ordinary differential equation is approximated by the implicit second order backward differentiation formula. The analytical solutions, in terms of the Mittag-Leffler function for the linear fractional equation and numerical solutions, obtained by the finite difference method for the nonlinear fractional equation, are used to evaluate the solutions of the original advection–diffusion equation. Finally, comparisons between numerical and exact solutions and the error estimates show that the proposed procedure has a high convergence precision.

PDE formulation of some SABR/LIBOR market models and its numerical solution with a sparse grid combination technique

SABR models have been used to incorporate stochastic volatility to LIBOR market models (LMM) in order to describe interest rate dynamics and price interest rate derivatives. From the numerical point of view, the pricing of derivatives with SABR/LIBOR market models (SABR/LMMs) is mainly carried out with Monte Carlo simulation. However, this approach could involve excessively long computational times. For first time in the literature, in the present paper we propose an alternative pricing based on partial differential equations (PDEs). Thus, we pose original PDE formulations associated to the SABR/LMMs proposed by Hagan and Lesniewsk (2008), Mercurio and Morini (2009) and Rebonato and White (2008). Moreover, as the PDEs associated to these SABR/LMMs are high dimensional in space, traditional full grid methods (like standard finite differences or finite elements) are not able to price derivatives over more than three or four underlying interest rates. In order to overcome this curse of dimensionality, a sparse grid combination technique is proposed. A comparison between Monte Carlo simulation results and the ones obtained with the sparse grid technique illustrates the performance of the method.

Numerical solution of modified differential equations based on symmetry preservation.

In this paper, we propose a method to construct invariant finite-difference schemes for solution of partial differential equations (PDEs) via consideration of modified forms of the underlying PDEs. The invariant schemes, which preserve Lie symmetries, are obtained based on the method of equivariant moving frames. While it is often difficult to construct invariant numerical schemes for PDEs due to complicated symmetry groups associated with cumbersome discrete variable transformations, we note that symmetries associated with more convenient transformations can often be obtained by appropriately modifying the original PDEs. In some cases, modifications to the original PDEs are also found to be useful in order to avoid trivial solutions that might arise from particular selections of moving frames. In our proposed method, modified forms of PDEs can be obtained either by addition of perturbation terms to the original PDEs or through defect correction procedures. These additional terms, whose primary purpose is to enable symmetries with more convenient transformations, are then removed from the system by considering moving frames for which these specific terms go to zero. Further, we explore selection of appropriate moving frames that result in improvement in accuracy of invariant numerical schemes based on modified PDEs. The proposed method is tested using the linear advection equation (in one- and two-dimensions) and the inviscid Burgers' equation. Results obtained for these tests cases indicate that numerical schemes derived from the proposed method perform significantly better than existing schemes not only by virtue of improvement in numerical accuracy but also due to preservation of qualitative properties or symmetries of the underlying differential equations.

Numerical simulation of the nuclear reactors kinetics behavior using method of lines based on the GDQ method

In this paper, for the first time, numerical technique using Method of Lines (MOL) based on the Generalized Differential Quadrature method (GDQ) is developed and applied for two energy groups of reactor kinetics and one group of precursor delayed neutrons. Also, the presented method (MOL-GDQ) has been developed for 3D transient benchmark nuclear reactor with two neutron energy groups and six delayed precursor groups. The basic idea of the MOL is to replace the spatial (boundary value) derivatives in the Partial Differential Equations (PDE) with algebraic approximations. In other words, with only one remaining independent variable, we have a system of Ordinary Differential Equations (ODEs) that approximates the original PDE. The advantages of the GDQ method lie in its easy use and flexibility with regard to arbitrary grid spacing. Compared to the conventional low-order numerical techniques such as the finite element and finite difference methods, the GDQ method can yield accurate solutions with relatively much fewer grid points. The numerical technique is applied to three-dimensional space-time neutron diffusion equations with average one group of delayed neutrons in the different nuclear reactors. Also, the presented method (MOL-GDQ) is developed for 3D transient benchmark nuclear reactor with two neutron energy groups and six delayed precursor groups. Simulation results are presented to demonstrate the effectiveness of the proposed method in terms of performance and stability. The results of numerical technique are discussed and compared with the results of traditional methods and strong citations to demonstrate the accuracy of it.

Adaptive multipreconditioned FETI: Scalability results and robustness assessment

The purpose of this article is to assess the adaptive multipreconditioned FETI solvers (AMPFETI) on real industrial problems and hardware. The multipreconditioned FETI algorithm (first introduced as Simultaneous FETI (Gosselet et al., 2015) is a non-overlapping domain decomposition method which exhibits good robustness properties without requiring the explicit knowledge of the original partial differential equation, or any a priori analysis of the algebraic system through eigenvalues problems. Multipreconditioned FETI solves critical problems in significantly fewer iterations than classical FETI but each iteration involves a larger computational effort. An adaptive strategy (known as the adaptive multipreconditioned conjugate gradient algorithm (Spillane, 2016) has been proposed to achieve balance between robustness and efficiency and we will observe that it provides an efficient solver for the problems considered here.

Simulation on Natural Convection of a Nanofluid along an Isothermal Inclined Plate

A numerical algorithm is presented for studying laminar natural convection flow of a nanofluid along an isothermal inclined plate. By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a set of nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab environment. Dimensionless velocity, temperature profiles and nanoparticle concentration for various angles of inclination are illustrated graphically. The effects of Prandtl number, Brownian motion parameter and thermophoresis parameter on Nusselt number are also discussed. The results of the present simulation are then compared with previous one available in literature with good agreement.

Analytical traveling-wave solutions to a generalized Gross–Pitaevskii equation with some new time and space varying nonlinearity coefficients and external fields

We present analytical matter-wave solutions to a generalized Gross–Pitaevskii (GGP) equation with several new time and space varying nonlinearity coefficients and external fields. This is realized by taking a suitable similarity transformation to the GGP equation which makes the original partial differential equation into a stationary and ordinary differential equation. We report a few families of analytical solutions of the GGP equation with several new time and space varying nonlinearity interactions, in which some physically relevant soliton solutions are found. The profile features of the evolution wave functions depend on the different choices of the composite functions ξ.

Intelligent computing approach to analyze the dynamics of wire coating with Oldroyd 8-constant fluid

In the study, intelligent computing technique is developed for solving the nonlinear system for wire coating analysis with the bath of Oldroyd 8-constant fluid having pressure gradient using feedforward artificial neural networks (ANNs), evolutionary computing, active-set algorithm (ASA) and their hybrid. Original partial differential equations of wire coating process are converted to nonlinear ordinary differential equation (NL-ODEs) in dimensionless form using similarity transformation. Strength of ANNs is exploited to develop mathematical model of the transformed equations by defining an unsupervised error. Training of design variables of the network is carried out globally using evolutionary computing techniques based on genetic algorithms (GAs) hybrid with ASA for rapid local convergence. Design scheme is applied to analyze the dynamics of the problem for number of variants based on dilatant constant, the pseudoplastic constant, the pressure gradient, shear stress under the effect of viscosity parameter and varying the coating thickness of the polymer. Results of the proposed method are compared with standard numerical solver for NL-ODEs based of Adams method to establish its correctness. Reliability of the method is further validated through the results of statistics based on different performance measures for accuracy and computational complexity.

Buckling analysis of thick plates on biparametric elastic foundation: a MAEM solution

The meshless analog equation method, a purely meshless method, is applied to the buckling analysis of moderately thick plates described by Mindlin’s theory and resting on two-parameter elastic foundation (Pasternak type). The method is based on the concept of the analog equation, which converts the three governing second-order partial differential equations (PDEs) in terms of the three plate displacements (transverse displacement and two rotations) into three substitute equations, the analog equations. The analog equations constitute a set of three uncoupled Poisson’s equations under fictitious sources, which are approximated by multi-quadric radial basis functions (MQ-RBFs) series. This enables the direct integration of the analog equations and allows the representation of the sought solution by new RBFs series. These RBFs approximate accurately not only the displacements but also their derivatives involved in the governing equations. Then, inserting the approximate solution in the original PDEs and in the associated boundary conditions (BCs) and collocating at mesh-free nodal points, a generalized eigenvalue problem is obtained, which allows the evaluation of the buckling load and the buckling modes. The studied examples demonstrate the efficiency of the presented method that is its ability to solve accurately and in a straightforward way difficult engineering problems.

Stochastic boundary control design for Timoshenko beams with large motions

This paper considers modeling and boundary control of Timoshenko beams with large motions under both deterministic and stochastic external loads. The original nonlinear partial differential equations governing motion of the beams are derived and used in the control design. The control design is based on the Lyapunov direct method. The proposed controllers guarantee globally practically K∞-exponentially p-stability of the beam motions at the reference state. Well-posedness and stability are analyzed based on a Lyapunov-type theorem developed to study well-posedness and stability for a class of stochastic evolution systems in Hilbert space. Simulation results are included to illustrate the effectiveness of the proposed control design.

Modeling and vibration control of a flexible aerial refueling hose with variable lengths and input constraint

In this paper, we present boundary control design for a flexible aerial refueling hose with varying length, varying speed, and input constraint. By the extended Hamilton’s principle, the flexible hose is modeled as a distributed parameter system described by partial differential equations (PDEs). Then a boundary control scheme is proposed based on the original PDEs to regulate the hose’s vibration and handle the effect of the input constraint. It is shown that the state of the system is proven to converge to a small neighborhood of zero in the presence of the varying length, varying speed and input constraint. The results are illustrated using numerical simulations for control performance verification.

Development of an upwinding kernel in SPH-SWEs model for 1D trans-critical open channel flows

In this study, an upwinding SPH model with a non-symmetric kernel function is proposed to predict one-dimensional open channel flows. Due to the application of non-symmetric kernel function biased in favor of the upstream side, numerical diffusion is intrinsically added into the discretized momentum equation using SPH. The proposed model thus has shown to have good potential to resolve steep gradient or discontinuous solutions without the need of exactly adding artificial viscosity to the discretized equation. Furthermore, an upwinding coefficient for the determination of the degree of upwinding is derived to accommodate the dispersion-relation-preserving (DRP) property. In wave number space, the error between the discretized SPH equations and the original partial differential equations is minimized, thereby yielding the optimized upwinding coefficient. The proposed model has been validated by solving four benchmark problems involving non-rectangular cross section, varying channel width, non-uniform bed slope and hydraulic jump. Comparison of the numerical and exact solutions shows that the proposed model has the ability of accurately predicting various open channel flows involving complicated transcritical flows. The consistency condition of the proposed model is also analyzed theoretically for the sake of completeness.

The onset of thermalization in finite-dimensional equations of hydrodynamics: insights from the Burgers equation

Solutions to finite-dimensional (all spatial Fourier modes set to zero beyond a finite wavenumber K G ), inviscid equations of hydrodynamics at long times are known to be at variance with those obtained for the original infinite dimensional partial differential equations or their viscous counterparts. Surprisingly, the solutions to such Galerkin-truncated equations develop sharp localized structures, called tygers (Ray et al. 2011 Phys. Rev. E 84 , 016301 ( doi:10.1103/PhysRevE.84.016301 )), which eventually lead to completely thermalized states associated with an equipartition energy spectrum. We now obtain, by using the analytically tractable Burgers equation, precise estimates, theoretically and via direct numerical simulations, of the time τ c at which thermalization is triggered and show that τ c ∼ K G ξ , with ξ = − 4 9 . Our results have several implications, including for the analyticity strip method, to numerically obtain evidence for or against blow-ups of the three-dimensional incompressible Euler equations.

Разрушение решения уравнения стратификации объемного заряда в полупроводниках: численный анализ при сведении исходного уравнения к дифференциально-алгебраической системе

Показана эффективность одной из методик численной диагностики разрушения решения на примере решения нелинейного уравнения cоболевского типа, описывающего стратификацию объемного заряда в полупроводниках. В процессе численного исследования этой задачи использовался подход, основанный на сведении исходного уравнения в частных производных к дифференциально-алгебраической системе с последующим решением этой системы с помощью одностадийной схемы Розенброка с комплексным коэффициентом. Численная диагностика разрушения точного решения указанного уравнения основывалась на методике вычисления апостериорной асимптотически точной оценки погрешности, получаемой при вычислении приближенного решения на последовательно сгущающихся сетках. The efficiency of one of the methods for the numerical diagnostics of solution's blow-up is shown using the example of solving a nonlinear Sobolev-type equation that describes a space charge stratification in semiconductors. An approach to reduce the original partial differential equation to a differential-algebraic system is used. This system is solved by the Rosenbrock scheme with a complex coefficient. The numerical diagnostics of solution's blow-up is based on the Richardson extrapolation procedure.

Moving Mesh Strategies of Adaptive Methods for Solving Nonlinear Partial Differential Equations

This paper proposes moving mesh strategies for the moving mesh methods when solving the nonlinear time dependent partial differential equations (PDEs). Firstly we analyse Huang’s moving mesh PDEs (MMPDEs) and observe that, after Euler discretion they could be taken as one step of the root searching iteration methods. We improve Huang’s MMPDE by adding one Lagrange speed term. The proposed moving mesh PDE could draw the mesh to equidistribution quickly and stably. The numerical algorithm for the coupled system of the original PDE and the moving mesh equation is proposed and the computational experiments are given to illustrate the validity of the new method.

Vibration control of a flexible aerial refuelling hose with input saturation

ABSTRACTIn this study, we consider a boundary control problem of a flexible aerial refuelling hose in the presence of input saturation. To provide an accurate and concise representation of the hose's behaviour, the flexible hose is modelled as a distributed parameter system described by partial differential equations (PDEs). By using the backstepping method, a boundary control scheme is proposed based on the original PDEs to regulate the hose's vibration. An auxiliary system based on a smooth hyperbolic function and a Nussbaum function is designed to handle the effect of the input saturation. Then based on Lyapunov's direct method, the state of the system is proven to converge to a small neighbourhood of zero by appropriately choosing design parameters. Finally, the results are illustrated using numerical simulations for control performance verification.

Dynamic optimization of trajectory for ramp‐up current profile in tokamak plasma

AbstractIn this paper, we consider an open‐loop, finite‐time, optimal control problem of attaining a specific desired current profile during the ramp‐up phase by finding the best open‐loop actuator input trajectories. Average density, total power, and plasma current are used as control actuators to manipulate the profile shape in tokamak plasmas. Based on the control parameterization method, we propose a numerical solution procedure directly to solve the original partial differential equation (PDE)‐constrained optimization problem using gradient‐based optimization techniques such as sequential quadratic programming. This paper is aimed at proposing an effective framework for the solution of PDE‐constrained optimization problem in tokamak plasmas. A more user‐friendly and efficient graphical user interface is designed in MATLAB, and the numerical simulation results are verified to demonstrate its applicability. In addition, the proposed framework of combining existing PDE and numerical optimization solvers to solve PDE‐constrained optimization problem has the prospective to target challenge advanced control problems arising in more general chemical engineering processes. © 2016 Curtin University of Technology and John Wiley & Sons, Ltd.