Numerical Solution Of Differential Equations Research Articles

Diagonalization of diffusion equations in two and three dimensions

Multidimensional diffusion is central to radiation belt and ring current dynamics, but obtaining numerical solutions reliably is subject to difficulties related to “cross terms” in the diffusion equation. Eliminating them, by constructing new coordinates which diagonalize the diffusion matrix, has been found to be effective in two dimensions. Here this approach is reformulated to be both clearer and more robust, with analytical expressions replacing numerical solutions of differential equations. An approach to extending the method to three dimensions is presented and discussed.

The basic principles of quartz radiofluorescence dynamics in the UV – analytical, numerical and experimental results

An analytical solution for the behaviour of quartz radiofluorescence (RF) in the UV-band is described based on a kinetic model involving one (deep) electron trap and two kinds of recombination centres. This model has been previously used to provide a qualitative description of quartz UV-RF. The derived numerical solution of differential equations describing charge transport in quartz can successfully reproduce experimental data. Here, this set of differential equations is solved analytically by assuming a dynamic balance during the RF stimulation. The analytical results are compared with numerical solutions and experimentally derived data. With the analytical solutions a better understanding of common natural quartz UV-RF behaviour is provided, and several experimentally observed phenomena can now be explained. Furthermore the comparison of two different kinetic models shows that the characteristic decay of the UV-RF signal in preheated quartz is attributed to an increasing competition of radiative and non-radiative centres during RF.

Numerical solution of differential equations arising in fluid dynamics using Legendre wavelet collocation method

Wavelet analysis is a recently developed mathematical tool for many problems. One of its main attractive features is the ability to accurately represent general functions with small number of adaptively chosen wavelet coefficients. In this paper, the Legendre wavelet collocation method (LWCM) for the numerical solution of differential equations is presented. The proposed method gives better results than the existing ones. Some of the illustrative examples are included to observe the performance of the proposed scheme.

RETRACTED ARTICLE: A review on the solution of Grad–Shafranov equation in the cylindrical coordinates based on the Chebyshev collocation technique

Equilibrium reconstruction consists of identifying, from experimental measurements, a distribution of the plasma current density that satisfies the pressure balance constraint. Numerous methods exist to solve the Grad–Shafranov equation, describing the equilibrium of plasma confined by an axisymmetric magnetic field. In this paper, we have proposed a new numerical solution to the Grad–Shafranov equation (an axisymmetric, magnetic field transformed in cylindrical coordinates solved with the Chebyshev collocation method) when the source term (current density function) on the right-hand side is linear. The Chebyshev collocation method is a method for computing highly accurate numerical solutions of differential equations. We describe a circular cross-section of the tokamak and present numerical result of magnetic surfaces on the IR-T1 tokamak and then compare the results with an analytical solution.

EXCELによる微分方程式の数値解を活用した力学の授業(私の工夫・私の実践)

EXCELによる微分方程式の数値解を活用した力学の授業(私の工夫・私の実践)

High order approximations using spline-based differential quadrature method: Implementation to the multi-dimensional PDEs

A new differential quadrature method based on cubic B-spline is developed for the numerical solution of differential equations. In order to develop the new approach, the B-spline basis functions are used on the grid and midpoints of a uniform partition. Some error bounds are obtained by help of cubic spline collocation, which show that the method in its classic form is second order convergent. In order to derive higher accuracy, high order perturbations of the problem are generated and applied to construct the numerical algorithm. A new fourth order method is developed for the numerical solution of systems of second order ordinary differential equations. By solving some test problems, the performance of the proposed methods is examined. Also the implementation of the method for multi-dimensional time dependent partial differential equations is presented. The stability of the proposed methods is examined via matrix analysis. To demonstrate the applicability of the algorithms, we solve the 2D and 3D coupled Burgers’ equations and 2D sine-Gordon equation as test problems. Also the coefficient matrix of the methods for multi-dimensional problems is described to analyze the stability.

Numerical Solution of Differential Equations by Direct Taylor Expansion

A variation of the direct Taylor expansion algorithm is suggested and applied to several linear and nonlinear differential equations of interest in physics and engineering, and the results are compared with those obtained from other algorithms. It is shown that the suggested algorithm competes strongly with other existing algorithms, both in accuracy and ease of application, while demanding a shorter computation time.

Numerical accuracy between Runge-Kutta Fehlberg method and Adams-Bashforth method for first order ordinary differential equations with boundary value

There are many problems in the field of science, engineering and technology which can be solved by differential equations formulation. This research will compare the accuracy of various method, the Runge-Kutta Fehlberg method and Adams-Bashforth method, in completing numerical solutions of differential equations, which is limited to ordinary differential equation of first order. Numbers of differential equations solved by the MATLAB version 7.9 to compare the accuracy between Runge-Kutta Fehlberg and Adams-Bashforth method. It can be concluded that Runge-Kutta Fehlberg method as more rigorous accuracy than the Adams-Bashforth method for solving linear ordinary differential equations of first order [1].

Numerical solution for the fluid flow between active elastic walls

We analyse a model of the fluid flow between elastic walls simulating arteries actively interacting with the blood. The lubrication theory for the flow is coupled with the pressure and shear stress from the walls. The resulting nonlinear partial differential equation describes the displacement of the walls as a function of the distance along the flow and time. The equation is solved numerically using the one-dimensional integrated radial basis function network method. A solution in the form of a self-sustained train of pulses is obtained. Numerical experiments demonstrate the process of formation of the train from randomly chosen initial conditions. Dependence of the pulses on the boundary conditions is explored. References S. J. Sherwin, L. Formaggia, J. Peiro and V. Franke. Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. Int. J. Numer. Meth. Fluids, 43:673–700, 2003. doi:10.1002/fld.543 C. Kleinstreuer. Biofluid Dynamics. Taylor and Francis, 2006. M. S. Olufsen, C. S. Peskin, W. Y. Kim, E. M. Pedersen, A. Nadim and J. Larsen. Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Eng., 28:1281–1299, 2000. doi:10.1114/1.1326031 K. S. Matthys, J. Alastruey, J. Peiro, A. W. Khir, P. Segers, P. R. Verdonck, K. H. Parker and S. J. Sherwin. Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements. J. Biomech., 40:3476–3486, 2007. doi:10.1016/j.jbiomech.2007.05.027 A. J. Roberts. A One-dimensional Introduction to Continuum Mechanics. World Scientific. Singapore, 1994. doi:10.1142/2496 D. V. Strunin, Fluid flow between active elastic plates. ANZIAM J., 50:C871–C883, 2009. doi:10.21914/anziamj.v50i0.1452 D. V. Strunin, Autosoliton model of the spinning fronts of reaction. IMA J. Appl. Math., 63:163–177, 1999. doi:10.1093/imamat/63.2.163 R. Huang and Z. Suo. Wrinkling of a compressed elastic film on a viscous layer. J. Appl. Phys., 91:1135–1142, 2002. doi:10.1063/1.1427407 J. King. The isolation oxidation of silicon: The reaction-controlled case. SIAM J. Appl. Math. 49:1064–1080, 1989. doi:10.1137/0149064 D. Ngo-Cong, N. Mai-Duy, W. Karunasena and T. Tran-Cong. Local moving least square-one-dimensional integrated radial basis function networks technique for incompressible viscous flows. Int. J. Numer. Meth. Fluids, 70:1443–1474, 2012. doi:10.1002/fld.3640 P. Le, N. Mai-Duy, T. Tran-Cong and G. Baker. A meshless modeling of dynamic strain localization in quasi-brittle materials using radial basis function networks, CMES–Comp. Model. Eng., 25:43–66, 2008. doi:10.3970/cmes.2008.025.043 N. Mai-Duy and T. Tran-Cong. Numerical solution of differential equations using multiquadric radial basis function networks, Neural Networks, 14:185–199, 2001. doi:10.1016/S0893-6080(00)00095-2

Numerical methods for the computation of the confluent and Gauss hypergeometric functions

The two most commonly used hypergeometric functions are the confluent hypergeometric function and the Gauss hypergeometric function. We review the available techniques for accurate, fast, and reliable computation of these two hypergeometric functions in different parameter and variable regimes. The methods that we investigate include Taylor and asymptotic series computations, Gauss---Jacobi quadrature, numerical solution of differential equations, recurrence relations, and others. We discuss the results of numerical experiments used to determine the best methods, in practice, for each parameter and variable regime considered. We provide roadmaps with our recommendation for which methods should be used in each situation.

Study on application of capillary plane radiation air conditioning system based on the slope roof

In this paper, based on the principle of the capillary plane radiation air conditioning system, taking the slope roof as an example, the application of the capillary plane radiation airconditioning system is studied and analysed. Then the numerical solution of differential equations is obtained by the technology of CFD. Finally, we analyze the distribution of indoor temperature of the slope roof and the predicted mean votes (PMV) using Airpak simulation software by establishing a physical model. The results show that the PMV of different sections ranges from 0 to 2.5, which meets the requirement of the comfort. These provide a theoretical basis for application and promotion of capillary plane in the slope roof.

Deformation modelling of structures enriched by inter-element continuity condition based on multi-sensor data fusion

A new adaptive method for deformation modelling of structures based on finite elements is developed under specified conditions in this paper. The continuity condition of deformation between overlaid surface elements on the structure is proposed to overcome the discontinuity of the surface deformation model. In the other words, this newly developed finite element method (DFEM) is designed by implementation of inter-element continuity conditions for a numerical solution of differential equations of the deformation model. Another capability of the proposed model is use of asynchronous and repetitious monitoring data of the structure (given by different geotechnical and geodetic sensors) as boundary values which enables fusion of multi-sensor measurements at data level in an inertial reference frame.Practical applicability of the method was assessed by successful determination of an earth-dam surface settlement variations based on the adaptive fusion of geodetic levelling and geotechnical settlement observations with inter-element continuity constraints. Regarding the observable continuity in the final surface deformation model contour maps also the estimated RMS (0.8mm) of difference between the deformation observations and their prediction on the check points, the presented method was more successful related to the deformation modelling method based on the single sensor data as well as the multi-sensor data fusion without inter-element continuity constraints.

Geometric Approximation: A New Computational Approach To Characterize Protein Dynamics from NMR Adiabatic Relaxation Dispersion Experiments.

A new computational strategy is reported that provides a fast approximation of numerical solutions of differential equations in general. The method is demonstrated with the analysis of NMR adiabatic relaxation dispersion experiments to reveal biomolecular dynamics. When an analytical solution to the theoretical equations describing a physical process is not available, the new approach can significantly accelerate the computational speed of the conventional numerical integration up to 10(5) times. NMR adiabatic relaxation dispersion experiments enhanced with optimized proton-decoupled pulse sequences, although extremely powerful, have previously been refractory to quantitative analysis. Both simulations and experimental validation demonstrate detectable "slow" (microsecond to millisecond) conformational exchange rates from 10(2) to 10(5) s(-1). This greatly expanded time-scale range enables the characterization of a wide array of conformational fluctuations for individual residues, which correlate with biomolecular function and were previously inaccessible. Moreover, the new computational method can be potentially generalized for analysis of new types of relaxation dispersion experiments to characterize the various dynamics of biomolecular systems.

Statistical analysis of differential equations: introducing probability measures on numerical solutions

In this paper, we present a formal quantification of uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite-dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it can be important to explicitly account for the uncertainty introduced by the numerical method. Doing so enables objective determination of this source of uncertainty, relative to other uncertainties, such as those caused by data contaminated with noise or model error induced by missing physical or inadequate descriptors. As ever larger scale mathematical models are being used in the sciences, often sacrificing complete resolution of the differential equation on the grids used, formally accounting for the uncertainty in the numerical method is becoming increasingly more important. This paper provides the formal means to incorporate this uncertainty in a statistical model and its subsequent analysis. We show that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of such differential equations. These measures exhibit contraction to a Dirac measure around the true unknown solution, where the rates of convergence are consistent with the underlying deterministic numerical method. Furthermore, we employ the method of modified equations to demonstrate enhanced rates of convergence to stochastic perturbations of the original deterministic problem. Ordinary differential equations and elliptic partial differential equations are used to illustrate the approach to quantify uncertainty in both the statistical analysis of the forward and inverse problems.

On the Exact Values of Daubechies Wavelets

In this paper we propose an algorithm for the calculation of the exact values of compactly-supported Daubechies wavelet functions. The algorithm is iterative, performing a single convolution operation at each step. It requires solving, at the first step only, a linear system of a relatively small size. The novelty of the algorithm is that once the values at dyadic points at a certain level j are calculated they do not need to be updated at the next step. We find that this algorithm is superior to the well-known cascade algorithm proposed by Ingrid Daubechies. This algorithm can serve well in wavelet based methods for the numerical solutions of differential equations. The algorithm is tested on Daubechies scaling functions as well as Daubechies coiflets. Comparison with the values obtained using the cascade algorithm is made. We found that the cascade algorithm results converge to ours

Simulation of Internal Stresses at the Metals and Alloys Radiation with Charged Particle Beams

The paper proposes a model for longitudinal and transverse stresses simulation by multigrid technique with numerical solutions of differential equations in partial derivatives for research the processes of detachment and cracking on the surface layers of metal and alloy products. The developed model can be useful at the optimal modes selection of surface treatment with electron and ion beams radiation. Simulation shows that in order to avoid the cracks appearance on the AISI A2 steel surface radiated with 400 keV electron beam energy the maximum radiation parameters should not exceed 0.9 kA/cm2 current density and 150ns pulse duration. This investigation assesses the stress arising at the matrix depth in a perpendicular direction to the electron beam and evaluates its magnitude that may exceed the stresses occurring through the matrix depth more than twice.

Diagnosis of Diabetic Disease with differential Equation using Fuzzy Convergence

Differential equations are highly useful in evaluating relationship between different physical quantities of scientific phenomena. In recent years, diagnosis and treatment of diabetes has been the subject of discussion in many research works with multiple physical measuring quantities (sugar level, insulin secretion, fat, carbohydrate, etc.,). Complexity of a human biological system allows its relations to be demonstrated in a nonlinear pattern. As the glucose levels in diabetic patients are a nonlinear model, differential equation using fuzzy controller is a good choice for controlling insulin in blood. In this paper, a method called Differential Equation using Fuzzy Logic System (DE-FLS) is presented to control the glucose level in insulin dependent diabetic patients. Formulates of third order differential equation for diabetes is investigated that helps in maintaining the glucose level. To improve the convergence rate, fuzzy logic system is designed to identify the relationships based on patient's data to numerical solutions of differential equation. It is shown that the proposed method, perform well in evaluating physical quantity relationship and its convergence rate gets improved with DE-FLS.

Research of stress-strain state and stability of a rokfill dam under seismic actions

One of the main factors determining the safety of earth sea and river hydraulic structures erected on water-saturated grounds is the process of consolidation, manifested under the action of static and seismic loads. A feature of cohesionless soils located in the structure itself or in its base, is their potential ability to liquefaction under seismic impacts. This paper describes the method of calculating the saturated soil’s environments under seismic actions based on the numerical solution of differential equations of the theory of consolidation by finite element method. The results of the static problem solving for the phased construction of the installation are used as the initial conditions. In order to describe the deformability of soil materials mathematical model formed by the theory of plastic flow with hardening is used. The parameters of this model are determined by the results of triaxial testing of soils. As an example, we study the interaction of a sea rockfill dam with a sandy base under seismic impacts, determined by the synthetic accelerograms. The results of calculations of the stress-strain state of the two sections of the dam (shallow and deep) are presented, and assessment is made of the possibility of liquefaction of sandy soil base. It is shown that the pore pressure that occurs in water-saturated cohesionless soil base and the body of the dam under seismic impacts, unloads the soil skeleton, which leads to a decrease in local shear safety factors. And, in the less dense soil base of the shallow section of the dam, the soil skeleton is unloaded to a greater extent, which negatively affects its overall safety factor.

Optimization of heat pump system

The purpose of this study is to produce a mathematical model to describe the operation of a water-to-water heat pump system for steady-state condition. The set-up model is deterministic. It consists of distributed as well as lumped parameters. The proposed mathematical models of heat exchangers were described by coupled differential equations, while the models of the compressor and the expansion valve are of lumped parameters. The Runge–Kutta and the Adams–Moulton predictor-corrector methods were applied for the numerical solution of differential equations, i.e. the equation systems. The developed mathematical model is validated with 118 tests using R134a as a working fluid. The results show that an average difference between the modeled and experimental results for the coefficient of performance is 1.73%, which means that the proposed mathematical model can be used to determine the optimum operating point of a heat pump system for a given heat demand for heating, by determining the maximum value of the coefficient of performance.

Analysis of the dengue risk by means of a Takagi–Sugeno-style model

Dengue is an infectious disease caused by an arbovirus (transmitted by the bite of the Aedes aegypti mosquito), which occurs mainly in tropical and subtropical areas, including Brazil. Nowadays it constitutes a major public health problems in the world. This paper proposes analysis of the evolution of dengue risk, from a model of the kind Takagi–Sugeno where the consequence of each fuzzy rule is a partial differential equation – PDE. The uncertainty parameters κi, related to the risk of spread of dengue, are directly linked to the behavior of the mosquito population and they were evaluated by making use of a system based on fuzzy rules and information provided by experts. These parameters depend on the population of humans, which provides blood to the maturation of eggs, and potential mosquito breeding containers and still depend on the rainfall. The amount of rainfall presents stochastic dependence in the sampled values, and for this reason we chose the Markov chain method to model it. Researchers at the Laboratory for Spatial Analysis of Epidemiological Data (epiGeo – UNICAMP) performed studies on the dengue risk in the southern region of the city of Campinas (São Paulo state, Brazil) between 2006 and 2007, by means of statistical methods and provided information to begin this study. The numerical solution of differential equation resulting from Takagi–Sugeno inference was obtained using the weighted essentially non-oscillatory schemes, the fifth order (WENO-5) not smooth for regions of the domain and centered finite difference schemes of high order for the smooth regions, in the space discretization. Also a lifting scheme was made to define smoothness in the regions. For the time evolution we have chosen Runge–Kutta TVD (Total Variation Diminishing), of the third order. Simulations corresponding to a period of summer (December, January and February) were performed and the results have showed that an effective action to reduce the risk of dengue is to severely combat the potential mosquito breeding sites.