Technique For Partial Differential Equations Research Articles

Single image de-hazing using adaptive dynamic stochastic resonance and wavelet-based fusion

This paper presents enhancement-based single image de-hazing algorithms using dynamic stochastic resonance (DSR) and wavelet fusion. The initial algorithm employs DSR controlled with image perceptual quality metrics to automatically guide the process in an adaptive way via intersecting curves. The extent of de-hazing is determined by a parameter dynamically computed from the relevant regions of the segmented hazy image. The second algorithm combines the strengths of the DSR-based algorithm, partial differential equation (PDE), fractional calculus (FC) and multi-scale filtering-based de-hazing techniques via wavelet fusion. The problems addressed include colour distortion, image darkening, halo effects, edge, noise and sky region over-enhancement. Results indicate that the proposed approaches yield comparable or better results than several de-hazing algorithms from the literature in terms of visual clarity, objective metrics and runtime.

An efficient analytical technique for fractional partial differential equations occurring in ion acoustic waves in plasma

In this work, we apply an efficient analytical algorithm namely homotopy perturbation Sumudu transform method (HPSTM) to find the exact and approximate solutions of linear and nonlinear time-fractional regularized long wave (RLW) equations. The RLW equations describe the nature of ion acoustic waves in plasma and shallow water waves in oceans. The derived results are very significant and imperative for explaining various physical phenomenons. The suggested method basically demonstrates how two efficient techniques, the Sumudu transform scheme and the homotopy perturbation technique can be integrated and applied to find exact and approximate solutions of linear and nonlinear time-fractional RLW equations. The nonlinear expressions can be simply managed by application of He's polynomials. The result shows that the HPSTM is very powerful, efficient, and simple and it eliminates the round-off errors. It has been observed that the proposed technique can be widely employed to examine other real world problems.

Hamilton-Jacobi equations for optimal control on networks with entry or exit costs

We consider an optimal control on networks in the spirit of the works of Achdou et al. [NoDEA Nonlinear Differ. Equ. Appl. 20 (2013) 413–445] and Imbert et al. [ESAIM: COCV 19 (2013) 129–166]. The main new feature is that there are entry (or exit) costs at the edges of the network leading to a possible discontinuous value function. We characterize the value function as the unique viscosity solution of a new Hamilton-Jacobi system. The uniqueness is a consequence of a comparison principle for which we give two different proofs, one with arguments from the theory of optimal control inspired by Achdou et al. [ESAIM: COCV 21 (2015) 876–899] and one based on partial differential equations techniques inspired by a recent work of Lions and Souganidis [Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (2016) 535–545].

Water waves with moving boundaries

The unified transform, also known as the Fokas method, provides a powerful methodology for studying boundary value problems. Employing this methodology, we analyse inviscid, irrotational, two-dimensional water waves in a bounded domain, and in particular we study the generation of waves by a moving piecewise horizontal bottom, as it occurs in tsunamis. We show that this problem is characterised by two equations which involve only first-order derivatives. It is argued that under the assumptions of ‘small amplitude waves’ but not of ‘long waves’, the above two equations can be treated numerically via a recently introduced numerical technique for elliptic partial differential equations in a polygonal domain. In the particular case that the moving bottom is horizontal and under the assumption of ‘small amplitude waves’, but not of ‘long waves’, these equations yield a non-local generalisation of the Boussinesq system. Furthermore, under the additional assumption of ‘long waves’ the above system yields a Boussinesq-type system, which however includes the effect of the moving boundary.

Convergence and stability analysis of heterogeneous time step coupling schemes for parabolic problems

We propose and rigorously analyze the subcycling method based on primal domain decomposition techniques for first-order transient partial differential equations. In time dependent problems, it can be computationally advantageous to use different time steps in different regions. Smaller time steps are used in regions of significant changes in the solution and larger time steps are prescribed in regions with nearly stationary response. Subcycling can efficiently reduce the total computational cost. Crucial to our approach is a nonstandard heterogeneous temporal discretization. We begin with the discretization in time by the asynchronous Rothe method, which, in essence, involves a backward finite difference scheme assuming different time steps (fine and large time steps) in different parts of the computational domain. The emphasis of the paper is on qualitative properties of the new numerical scheme, such as a-priori estimates, existence of the time-discrete solutions and the strong convergence and stability analysis. Several numerical experiments were conducted to examine the consistency of the proposed method.

A Numerical Approximation Structured by Mixed Finite Element and Upwind Fractional Step Difference for Semiconductor Device with Heat Conduction and Its Numerical Analysis

AbstractA coupled mathematical system of four quasi-linear partial differential equations and the initial-boundary value conditions is presented to interpret transient behavior of three dimensional semiconductor device with heat conduction. The electric potential is defined by an elliptic equation, the electron and hole concentrations are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite element approximation is used to get the electric field potential and one order of computational accuracy is improved. Two concentration equations and the heat conduction equation are solved by a fractional step scheme modified by a second-order upwind difference method, which can overcome numerical oscillation, dispersion and computational complexity. This changes the computation of a three dimensional problem into three successive computations of one-dimensional problem where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by prior estimate theory and other special techniques of partial differential equations. This type of parallel method is important in numerical analysis and is most valuable in numerical application of semiconductor device and it can successfully solve this international famous problem.

Improved wavelets based technique for nonlinear partial differential equations

One of the most challenging task now a days for engineers and scientists is finding solutions of nonlinear Partial Differential Equations (PDEs) which frequently arise in many engineering and physical phenomena’s. Encouraged by the ongoing research, a new technique is proposed in this article for obtaining more accurate results of nonlinear PDEs. Shifted Legendre wavelets and Picard’s Iteration Technique are used in the proposed technique. To test the significance of the proposed technique, nonlinear Gardner equation is considered and solved. The proposed technique provides very accurate results over a wider interval because of the use of the shifted polynomials. The results obtained are also compared with the results of Variational Iteration Method and the supremacy of the proposed method is established.

Partial differential equation techniques for analysing animal movement: A comparison of different methods

Recent advances in animal tracking have allowed us to uncover the drivers of movement in unprecedented detail. This has enabled modellers to construct ever more realistic models of animal movement, which aid in uncovering detailed patterns of space use in animal populations. Partial differential equations (PDEs) provide a popular tool for mathematically analysing such models. However, their construction often relies on simplifying assumptions which may greatly affect the model outcomes. Here, we analyse the effect of various PDE approximations on the analysis of some simple movement models, including a biased random walk, central-place foraging processes and movement in heterogeneous landscapes. Perhaps the most commonly-used PDE method dates back to a seminal paper of Patlak from 1953. However, our results show that this can be a very poor approximation in even quite simple models. On the other hand, more recent methods, based on transport equation formalisms, can provide more accurate results, as long as the kernel describing the animal's movement is sufficiently smooth. When the movement kernel is not smooth, we show that both the older and newer methods can lead to quantitatively misleading results. Our detailed analysis will aid future researchers in the appropriate choice of PDE approximation for analysing models of animal movement.

Special Section: 2016 Copper Mountain Conference

The Fourteenth Copper Mountain Conference on Iterative Methods was held March 20--25, 2016, in Copper Mountain, Colorado. The meeting featured more than 170 presentations covering a broad spectrum of topics in scientific computing. Highlighted topics for the 2016 conference included iterative linear algebraic techniques in data mining, optimization, inverse problems, nonlinear solvers, methods for large-scale eigenvalue and singular value computations, implementation of iterative solvers on emerging architectures, uncertainty quantification, robust and scalable iterative solution of coupled multi-physics problems, model reduction, and methods based on low-rank approximations. Besides data mining, applications to energy, climate, imaging, fluids and electromagnetics were especially encouraged. Several speakers addressed discretization techniques for partial differential equations from the point of view of their impact on the design and performance of iterative solvers. As is customary for Copper Mountain, ...

Some properties for Itô processes driven by $G$-Brownian motion

In this paper, we use partial differential equation (PDE) techniques and probabilistic approaches to study the lower capacity of the ball for the Itô process driven by $G$-Brownian motion ($G$-Itô process). In particular, the lower bound on the lower capacity of certain balls is obtained. As an application, we prove a strict comparison theorem in $G$-expectation framework.

Numerical method of mixed finite volume-modified upwind fractional step difference for three-dimensional semiconductor device transient behavior problems

Transient behavior of three-dimensional semiconductor device with heat conduction is described by a coupled mathematical system of four quasi-linear partial differential equations with initial-boundary value conditions. The electric potential is defined by an elliptic equation and it appears in the following three equations via the electric field intensity. The electron concentration and the hole concentration are determined by convection-dominated diffusion equations and the temperature is interpreted by a heat conduction equation. A mixed finite volume element approximation, keeping physical conservation law, is used to get numerical values of the electric potential and the accuracy is improved one order. Two concentrations and the heat conduction are computed by a fractional step method combined with second-order upwind differences. This method can overcome numerical oscillation, dispersion and decreases computational complexity. Then a three-dimensional problem is solved by computing three successive one-dimensional problems where the method of speedup is used and the computational work is greatly shortened. An optimal second-order error estimate in L2 norm is derived by using prior estimate theory and other special techniques of partial differential equations. This type of mass-conservative parallel method is important and is most valuable in numerical analysis and application of semiconductor device.

ON APPROXIMATE AND CLOSED-FORM SOLUTION METHOD FOR INITIAL-VALUE WAVE-LIKE MODELS

This work presents a proposed Modified Differential Transform Method (MDTM) for obtaining both closed-form and approximate solutions of initial-value wave-like models with variable, and constant coefficients. Our results when compared with the exact solutions of the associated solved problems, show that the method is simple, effective and reliable. The results are very much in line with their exact forms. The method involves less computational work without neglecting accuracy. We recommend this simple proposed technique for solving both linear and nonlinear partial differential equations (PDEs) in other aspects of pure and applied sciences.

An irreversible investment problem with maintenance expenditure on a finite horizon: Free boundary analysis

This paper concerns a continuous-time, finite horizon, optimal irreversible investment problem with maintenance expenditure of a firm under uncertainty. We assume that the firm can make irreversible investments to expand its production capacity and spend maintenance expenditure to achieve better performance of the productivity. The objective of the firm is to construct optimal investment and maintenance policies to maximize its expected total profit over a finite horizon. Mathematically, it is a singular stochastic control problem whose value function satisfies a parabolic variational inequality with gradient constraint. The problem gives rise to two free boundaries which stand for the optimal investment and maintenance strategies, respectively. We investigate behaviors of free boundaries, study regularities of the value function, and give optimal investment and maintenance policies. As we know, this is a first integral result for an investment–maintenance problem with a finite time horizon due to use of partial differential equation (PDE) technique.

Numerical study of double diffusive buoyancy forces induced natural convection in a trapezoidal enclosure partially heated from the right sidewall

A study of double-diffusive natural convection in a trapezoidal enclosure with a partial heated active right sidewall has been conducted numerically using the finite difference method. The length of the heated active part is equal to half of the inclined wall. Uniform different temperatures and concentrations are imposed along the active parts of the enclosure. The top and bottom boundaries of the enclosure, as well as inactive part of the right sidewall, are being insulated and impermeable. The species diffusivity of the fluid is considered to be constant, but the density varies linearly with the temperature and concentration. Double-diffusive convection for laminar two-dimensional incompressible flow with negligible radiation is expressed in terms of vorticity, temperature or energy, concentration and stream function. A Partial Differential Equation (PDE) technique is adopted to generate regular grid distribution in the physical space. The numerical results are reported for the effect of different heating cases, thermal Grashof numbers, and inclination angles on the contours of streamline, temperature, and concentration. Also, the relevant results for the average Nusselt and Sherwood numbers are demonstrated for several parameters including thermal Grashof number (103⩽GrT⩽106), Lewis number (0.5⩽Le⩽10), Prandtl number (0.7⩽Pr⩽10) at a fixed aspect ratio Ar=1 and buoyancy ratio N=-0.2.

Domain Decomposition Modified with Characteristic Finite Element Method for Numerical Simulation of Semiconductor Transient Problem of Heat Conduction

A characteristic finite element algorithm based on domain decomposition is structured in this paper to approximate numerically multi-dimensional semiconductor transient problems of heat conduction. Finite element approximation is presented for the electric field potential equation, and a domain decomposition discretization with characteristic finite element is put forward for the electron concentration equation, hole concentration equation and heat conductor equation. An optimal order error estimate in L2 norm is derived for the coupled system by using some techniques such as variation, domain decomposition, the method of characteristics, the principle of energy, negative norm estimates, induction hypothesis, prior estimates theory and other techniques of partial differential equations. Finally, experimental data consistent with theoretical convergence rate are shown. This type of numerical method is of high computational efficiency and can successfully solve this international problem.

A numerical technique for linear elliptic partial differential equations in polygonal domains.

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform (or the Fokas transform) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map. The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

A semianalytical technique for solving nonlinear partial differential equations

A semianalytical technique is presented using Maple to solve coupled nonlinear partial differential equations PDEs. The technique is based on analytically solving the equations resulting from discretizing the spatial coordinate of the PDEs and using quasilinearization with Newton's method to treat nonlinear terms. © 1996 John Wiley & Sons, Inc.

Nodal sets of thin curved layers

This paper is concerned with the location of nodal sets of eigenfunctions of the Dirichlet Laplacian in thin tubular neighbourhoods of hypersurfaces of the Euclidean space of arbitrary dimension. In the limit when the radius of the neighbourhood tends to zero, it is known that spectral properties of the Laplacian are approximated well by an effective Schrödinger operator on the hypersurface with a potential expressed solely in terms of principal curvatures. By applying techniques of elliptic partial differential equations, we strengthen the known perturbation results to get a convergence of eigenfunctions in Hölder spaces. This enables us in particular to conclude that every nodal set has a non-empty intersection with the boundary of the tubular neighbourhood.

Pseudophysical Compartment Modeling of an Industrial Rotary Dryer with Flighted and Unflighted Sections: Solids Transport

A novel pseudophysical compartment model describing the solids transport within a case-study cocurrent industrial rotary dryer is presented. The model is novel and distinctive because it combines both multiscale partial differential equation and ordinary differential equation modeling techniques to represent the five separate sections occurring within the dryer, including both flighted and unflighted sections. The solid phase in the unflighted sections was modeled as an axial dispersed plug flow system. The solid transport in the flighted sections was partitioned into the usual series–parallel formulation of well-mixed tanks representing airborne and flight-borne solids. Parameter values and compartment numbers were estimated using mechanistic geometric modeling, dryer design loading constraints, and solids flow properties. Geometric modeling enabled the effects of internal scaling arising from solid materials sticking to the internal walls of the dryer to be accounted for in a realistic manner. Industria...

Reduced-order modelling numerical homogenization.

A general framework to combine numerical homogenization and reduced-order modelling techniques for partial differential equations (PDEs) with multiple scales is described. Numerical homogenization methods are usually efficient to approximate the effective solution of PDEs with multiple scales. However, classical numerical homogenization techniques require the numerical solution of a large number of so-called microproblems to approximate the effective data at selected grid points of the computational domain. Such computations become particularly expensive for high-dimensional, time-dependent or nonlinear problems. In this paper, we explain how numerical homogenization method can benefit from reduced-order modelling techniques that allow one to identify offline and online computational procedures. The effective data are only computed accurately at a carefully selected number of grid points (offline stage) appropriately 'interpolated' in the online stage resulting in an online cost comparable to that of a single-scale solver. The methodology is presented for a class of PDEs with multiple scales, including elliptic, parabolic, wave and nonlinear problems. Numerical examples, including wave propagation in inhomogeneous media and solute transport in unsaturated porous media, illustrate the proposed method.