Three-dimensional Partial Differential Equations Research Articles

Haar wavelet collocation method for three-dimensional elliptic partial differential equations

A new collocation method based on Haar wavelet is presented for numerical solution of three-dimensional elliptic partial differential equations with Dirichlet boundary conditions. An important advantage of the method is that it can be applied to both linear as well as nonlinear problems. The algorithm based on this new method is simple and can be easily implemented in any programming language. Experimental rates of convergence of the proposed method are calculated which are in agreement with theoretical results. The proposed method is applied to several benchmark problems from the literature including linear and nonlinear elliptic problems as well as systems of elliptic partial differential equations. The numerical experiments confirm the accuracy and diverse applicability of the method.

Mathematical model of parallel-plate-channeled electromagnetic-driven pulsating dream pipe for rapid heat removal

In response to technical claims that troubled nuclear reactors be safely cooled in a short period, this study discovers new merits in the so-called dream pipe which can be a kind of forced oscillatory heat pipe. Introduced are not capillary-tube-bundled dream pipes with sinusoidally vibrating mechanical shakers but parallel-plate-channeled ones with electromagnetic-driven pulse-width-modulating oscillators. Governing momentum-and-energy equations thus become quite different from those used in traditional dream pipes. An extended variational technique based on the Laplace-transform Ritz method is applied to solve the non-steady, three-dimensional partial differential equations. Possible velocity-and-temperature profiles are derived to find the effective thermal diffusivity and the tidal displacement, which are combined to yield two performance indexes for the convenience. Those two are theoretically and numerically compared with the analysis results of Watson and of Kurzweg to demonstrate that ours are acceptable. Numerical computations are made to graphically show the heat diffusive capacity of the proposed new-style dream pipes and then to present design specifications for them.

Numerical analyses of steady-state seepage problems using the interpolation finite difference method

Seepage analyses have mainly been executed using the finite element method; numerical analyses using the finite difference method (FDM) have been limited to cases where the calculation domains are comparatively simple. This limitation is observed because FDM is considered to be inappropriate for application in calculations over complex domains. However, by applying the so-called “interpolation FDM (IFDM)”, we can now freely solve two- and three-dimensional elliptic partial differential equations (PDEs) over complex domains with high speed and high accuracy. By adopting this procedure, named the boundary polynomial interpolation, all of the numerical analyses of elliptic PDEs reduce to Dirichlet problems over regular domains. This method is also effective in the calculation of a flow net where mixed Dirichlet and Neumann conditions exist. By giving the coordinate values of changing points regarding the polygonal line of a domain and boundary conditions, grid generation is automatically carried out and numerical solutions are promptly obtained. In this paper, the method of saturated seepage analyses with a fixed domain is first formulated and then expanded to unconfined domain problems, namely, free surface problems. While analytical solutions of the PDE are highly limited, there is an analytical solution for the location of the free surface in a rectangular dam. The numerical solutions obtained using the IFDM are compared with the analytical ones, and it is shown that the proposed method has adequate accuracy in practice and wide applicability as a general method of numerically solving seepage problems.

A two-dimensional Haar wavelets method for solving systems of PDEs

In this paper, we modify the idea of Haar wavelets method to obtain semi-analytical solutions for the systems of three-dimensional nonlinear partial differential equations. Theoretical considerations are discussed. To demonstrate the efficiency of the method, a test problem is presented. The approximate solutions of the system of three-dimensional nonlinear partial differential equations are compared with the exact solutions as well as existing numerical solutions found in the literature. The numerical solutions which are obtained using the suggested method show that numerical solutions are in a very good coincidence with the exact solutions.

On One Class of Quaternionic Mappings

We consider a new class of quaternionic mappings associated with three-dimensional partial differential equations and propose a description of all mappings from this class by using four analytic functions of complex variable.

Modeling and numerical calculation of three-dimensional non-steady state thermal explosion model of cylindrical battery

In order to study the thermal safety of cylindrical battery deeply, based on the theory of heat transfer, thermal explosion and nonlinear modeling, a three-dimensional non-steady state thermal explosion mathematical model of cylindrical fireworks with non-uniform heat dissipation of the lateral surface was established for the first time (three-dimensional partial differential equation group). Combining seven point difference method and Newton-homotopy algorithm, the numerical calculation method of the three-dimensional non-steady state thermal explosion partial differential equation was established and the numerical calculation program was written base on Matlab. The validity of calculation program has been demonstrated by comparison of numerical solutions and classical solutions. The accuracy of model has been validated by example computation and analysis. The critical parameters describing non-steady state model of cylindrical fireworks when stored individually and stored in combination form were calculated in this paper, including temperature distribution, temperature–time history, thermal explosion time to ignition, etc. The results show that when the ambient temperature is 450K, the fireworks stored individually do not have thermal explosion, but the fireworks stored in combination form will explode finally and the thermal explosion time to ignition is 19013.53s. When the ambient temperature is 460K, thermal explosion will occur in both the fireworks stored individually and stored in combination form, and the thermal explosion time to ignition are 3187.07s and 3066.60s respectively. It shows a more exact analytical methods and solutions of thermal safety evaluation of fireworks was established in this paper. Thus, it is need to strengthen the safety monitoring and management of cylindrical battery (combined fireworks) because of the higher thermal hazard.

Analysis of the Performance of Thermoelectric Modules Under Concentrated Radiation Heat Flux

The concentration of solar radiation by either a lens or a mirror is one of the options for practical utilisation of light to obtain higher temperatures. However, it is difficult to maintain high temperatures on the hot side of the module due to solar diurnal motion. This study evaluates the influence of the thermoelectric (TE) output by optical light concentration. Three-dimensional partial differential equations describing heat balance and TE phenomena were simultaneously solved by applying numerical methods, and the temperature distribution in the whole TE module as well as the current density were simulated. It was shown that the three models of light concentration on a single TE module (BiTe-based, four legs having dimensions of 10 mm × 10 mm × 10 mm) generate a similar output in the external load. This happens because the long leg becomes a large thermal resistance, and because the alumina plate (1 mm thick) with a high thermal conductivity covers the top of the TE modules. The homogenised temperature at the hot junctions generates a similar output in all three models when the cold terminals were kept at constant temperature.

Decomposition methods for coupled 3D equations of applied mathematics and continuum mechanics: Partial survey, classification, new results, and generalizations

The present paper provides a systematic treatment of various decomposition methods for linear (and some model nonlinear) systems of coupled three-dimensional partial differential equations of a fairly general form. Special cases of the systems considered are commonly used in applied mathematics, continuum mechanics, and physics. The methods in question are based on the decomposition (splitting) of a system of equations into a few simpler subsystems or independent equations. We show that in the absence of mass forces the solution of the system of four three-dimensional stationary and nonstationary equations considered can be expressed via solutions of three independent equations (two of which having a similar form) in a number of ways. The notion of decomposition order is introduced. Various decomposition methods of the first, second, and higher orders are described. To illustrate the capabilities of the methods, more than fifteen distinct systems of coupled 3D equations are discussed which describe viscoelastic incompressible fluids, compressible barotropic fluids, thermoelasticity, thermoviscoelasticity, electromagnetic fields, etc. The results obtained may be useful when constructing exact and numerical solutions of linear problems in continuum mechanics and physics as well as when testing numerical and approximate methods for linear and some nonlinear problems.

A boundary value approach for solving three-dimensional elliptic and hyperbolic partial differential equations.

In this article, the boundary value method is applied to solve three dimensional elliptic and hyperbolic partial differential equations. The partial derivatives with respect to two of the spatial variables (y, z) are discretized using finite difference approximations to obtain a large system of ordinary differential equations (ODEs) in the third spatial variable (x). Using interpolation and collocation techniques, a continuous scheme is developed and used to obtain discrete methods which are applied via the Block unification approach to obtain approximations to the resulting large system of ODEs. Several test problems are investigated to elucidate the solution process.

New analytical approach for transition to slow 3-D turbulence

Analytical non-perturbative study of the three-dimensional nonlinear stochastic partial differential equation with additive thermal noise analogous to that proposed by Nikolaevskii V.N. to describe longitudinal seismic waves is presented. The equation has a threshold of short-wave instability and symmetry providing long wave dynamics. New mechanism of quantum chaos generating in nonlinear dynamical systems with infinite number of degrees of freedom is proposed. The hypothesis says that physical turbulence could be identified with quantum chaos of considered type. It is shown that the additive thermal noise destabilizes dramatically the ground state of the Nikolaevskii system causing it to make a direct transition from a spatially uniform to a turbulent state.

Optimization of mesh hierarchies in multilevel Monte Carlo samplers

We perform a general optimization of the parameters in the Multilevel Monte Carlo (MLMC) discretization hierarchy based on uniform discretization methods with general approximation orders and computational costs. We optimize hierarchies with geometric and non-geometric sequences of mesh sizes and show that geometric hierarchies, when optimized, are nearly optimal and have the same asymptotic computational complexity as non-geometric optimal hierarchies. We discuss how enforcing constraints on parameters of MLMC hierarchies affects the optimality of these hierarchies. These constraints include an upper and a lower bound on the mesh size or enforcing that the number of samples and the number of discretization elements are integers. We also discuss the optimal tolerance splitting between the bias and the statistical error contributions and its asymptotic behavior. To provide numerical grounds for our theoretical results, we apply these optimized hierarchies together with the Continuation MLMC Algorithm. The first example considers a three-dimensional elliptic partial differential equation with random inputs. Its space discretization is based on continuous piecewise trilinear finite elements and the corresponding linear system is solved by either a direct or an iterative solver. The second example considers a one-dimensional It\^o stochastic differential equation discretized by a Milstein scheme.

Many-body microhydrodynamics of colloidal particles with active boundary layers

Colloidal particles with active boundary layers—regions surrounding the particles where non-equilibrium processes produce large velocity gradients—are common in many physical, chemical and biological contexts. The velocity or stress at the edge of the boundary layer determines the exterior fluid flow and, hence, the many-body interparticle hydrodynamic interaction. Here, we present a method to compute the many-body hydrodynamic interaction between N spherical active particles induced by their exterior microhydrodynamic flow. First, we use a boundary integral representation of the Stokes equation to eliminate bulk fluid degrees of freedom. Then, we expand the boundary velocities and tractions of the integral representation in an infinite-dimensional basis of tensorial spherical harmonics and, on enforcing boundary conditions in a weak sense on the surface of each particle, obtain a system of linear algebraic equations for the unknown expansion coefficients. The truncation of the infinite series, fixed by the degree of accuracy required, yields a finite linear system that can be solved accurately and efficiently by iterative methods. The solution linearly relates the unknown rigid body motion to the known values of the expansion coefficients, motivating the introduction of propulsion matrices. These matrices completely characterize hydrodynamic interactions in active suspensions just as mobility matrices completely characterize hydrodynamic interactions in passive suspensions. The reduction in the dimensionality of the problem, from a three-dimensional partial differential equation to a two-dimensional integral equation, allows for dynamic simulations of hundreds of thousands of active particles on multi-core computational architectures. In our simulation of 104 active colloidal particle in a harmonic trap, we find that the necessary and sufficient ingredients to obtain steady-state convective currents, the so-called ‘self-assembled pump’, are (a) one-body self-propulsion and (b) two-body rotation from the vorticity of the Stokeslet induced in the trap.

New Scenario for Transition to Slow 3-D Turbulence

Analytical non-perturbative study of the three-dimensional nonlinear stochastic partial differential equation with additive thermal noise, analogous to that proposed by V.N. Nikolaevskii [1]-[5]to describe longitudinal seismic waves, is presented. The equation has a threshold of short-wave instability and symmetry, providing long wave dynamics. New mechanism of quantum chaos generating in nonlinear dynamical systems with infinite number of degrees of freedom is proposed. The hypothesis is said, that physical turbulence could be identified with quantum chaos of considered type. It is shown that the additive thermal noise destabilizes dramatically the ground state of the Nikolaevskii system thus causing it to make a direct transition from a spatially uniform to a turbulent state.

Heat Transfer during Steaming of Bread

Abstract Understanding of heat transfer during steaming is important to optimize the processing of steamed bread and to produce desired qualities in the final product. Physicochemical changes occur during steaming of the dough which might be impacted upon by the heat transfer system. In this study, a mathematical model was developed to describe the heat transfer system in the bread being steaming throughout the heating process. The Forward Euler method was employed for solving the three-dimensional partial differential equation set for heat transfer to produce temperature profiles at a number of individual locations in the steamed bread during its steaming. All the comparisons between the model-predicted values and the experimental results produced root mean square error values ranged from 1.391 to 3.545 and R 2 values of all greater than 0.93. Therefore, it is confirmed that the model has a good performance and can be used to predict temperature profiles in the bread during steaming.

Efficient Numerical Algorithms for Three-Dimensional Fractional Partial Differential Equations

This paper detailedly discusses the locally one-dimensional numerical methods for efficiently solving the three-dimensional fractional partial differential equations, including fractional advection diffusion equation and Riesz fractional diffusion equation. The second order finite difference scheme is used to discretize the space fractional derivative and the Crank-Nicolson procedure to the time derivative. We theoretically prove and numerically verify that the presented numerical methods are unconditionally stable and second order convergent in both space and time directions. In particular, for the Riesz fractional diffusion equation, the idea of reducing the splitting error is used to further improve the algorithm, and the unconditional stability and convergency are also strictly proved and numerically verified for the improved scheme.

Numerical simulation of a new two-dimensional variable-order fractional percolation equation in non-homogeneous porous media

The inherent heterogeneities of many geophysical systems often give rise to fast and slow pathways to water and chemical movement. One approach to model solute transport through such media is by fractional diffusion equations with space–time dependent variable coefficients. Many physical processes appear to exhibit fractional-order behavior that may vary with time, or space, or space and time. The theory of pseudodifferential operators and equations has been used to deal with this situation. In this paper we use a fractional Darcy’s law with variable order Riemann–Liouville fractional derivatives, this leads to a new variable-order fractional diffusion equation with variable coefficients.In this paper we consider a new two-dimensional variable-order fractional percolation equation with variable coefficients. An alternating direct method for the two-dimensional variable-order fractional percolation equation is proposed. Stability and convergence of the implicit alternating direct method are discussed. Finally, some numerical results are given. The numerical results demonstrate the effectiveness of the methods. These techniques can be used to simulate three-dimensional variable-order fractional partial differential equations.

Three-Dimensional Quadratic Model of Paclitaxel Release from Biodegradable Polymer Films

This paper presents a three-dimensional (3D) quadratic partial differential equation model of the drug release from a thin film of biodegradable polymer to a surrounding medium. Its very innovative feature is to go directly from the experimental normalized release curves of Lao and Venkatraman [J. Control. Release, 130 (2008), pp. 9--14] to a flux condition at the interface between the polymer and the medium that only requires the identification of the two parameters of the highly accurate ordinary differential equation model of Blanchet, Delfour, and Garon [SIAM J. Appl. Math., 71 (2011), pp. 2269--2286]. In the context of drug eluting stents, it is a practical and economical tool to theoretically and numerically simulate the 3D release of drug from the thin polymer film to the integrated wall and lumen of the blood vessel for evaluation and design. This approach avoids resorting to time-dependent or nonlinear diffusion in the polymer.

Rheological effects in the 3D creeping flow past a sedimenting sphere subject to orthogonal shear

The effects of the rheological parameters of nonlinear differential constitutive models in the isothermal, steady, creeping, flow past a sedimenting sphere, in an incompressible viscoelastic matrix fluid, subject to simple shear in a plane perpendicular to the direction of sedimentation are studied analytically. The viscoelasticity of the ambient fluid is modeled using the Upper Convected Maxwell, Oldroyd-B, exponential Phan-Thien-Tanner, Giesekus, and FENE-P constitutive equations. The solution of the governing equations is expanded as a regular perturbation series for small values of the Deborah number, and the resulting sequence of three-dimensional partial differential equations is solved analytically up to fourth order. Approximate analytical expressions for the angular velocity of the rigid sphere, as well as for the drag force on the sphere, are derived and discussed. The solutions reveal both the increase of the drag in case of Boger-type fluids (modeled with the FENE-P model) and the decrease of the drag in case of elastic fluids (modeled with the Giesekus model).

Series expansion solutions for the multi-term time and space fractional partial differential equations in two- and three-dimensions

Fractional partial differential equations with more than one fractional derivative in time describe some important physical phenomena, such as the telegraph equation, the power law wave equation, or the Szabo wave equation. In this paper, we consider two- and three-dimensional multi-term time and space fractional partial differential equations. The multi-term time-fractional derivative is defined in the Caputo sense, whose order belongs to the interval (1,2],(2,3],(3,4] or (0,m], and the space-fractional derivative is referred to as the fractional Laplacian form. We derive series expansion solutions based on a spectral representation of the Laplacian operator on a bounded region. Some applications are given for the two- and three-dimensional telegraph equation, power law wave equation and Szabo wave equation.

An Efficient Parallel Computing Method for the Processing of Large Sensed Data

In recent years we witness the advent of the Internet of Things and the wide deployment of sensors in many applications for collecting and aggregating data. Efficient techniques are required to analyze these massive data for supporting intelligent decisions making. Partial differential problems which involve large data are the most common in the engineering and scientific research. For simulations of large-scale three-dimensional partial differential equations, the intensive computation ability and large amounts of memory requirements for modeling are the main research problems. To address the two challenges, this paper provided an effective parallel method for partial differential equations. The proposed approach combines the overlapping domain decomposition strategy and the multi-core cluster technology to achieve parallel simulations of partial differential equations, uses the finite difference method to discretize equations and adopts the hybrid MPI/OpenMP programming model to exploit two-level parallelism on a multi-core cluster. The three-dimensional groundwater flow model with the parallel finite difference overlapping domain decomposition strategy was successfully set up and carried out by the parallel MPI/OpenMP implementation on a multi-core cluster with two nodes. The experimental results show that the proposed parallel approach can efficiently simulate partial differential problems with large amounts of data.