Search results for: Rosseland diffusion approximation

Authors: Hassan Al Salman

Abstract:

We consider a nonlinear parabolic cross diffusion model arising in applied mathematics. A fully practical piecewise linear finite element approximation of the model is studied. By using entropy-type inequalities and compactness arguments, existence of a global weak solution is proved. Providing further regularity of the solution of the model, some uniqueness results and error estimates are established. Finally, some numerical experiments are performed.

Keywords: cross diffusion model, entropy-type inequality, finite element approximation, numerical analysis

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Authors: Tudor Barbu

Abstract:

We present a robust nonlinear parabolic partial differential equation (PDE)-based denoising scheme in this article. Our approach is based on a second-order anisotropic diffusion model that is described first. Then, a consistent and explicit numerical approximation algorithm is constructed for this continuous model by using the finite-difference method. Finally, our restoration experiments and method comparison, which prove the effectiveness of this proposed technique, are discussed in this paper.

Keywords: anisotropic diffusion, finite differences, image denoising and restoration, nonlinear PDE model, anisotropic diffusion, numerical approximation schemes

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Authors: Hassan J. Al Salman, Ahmed A. Al Ghafli

Abstract:

In this study, we consider a nonlinear in time finite element approximation of a reaction diffusion system of lambda-omega type. We use a fixed-point theorem to prove existence of the approximations at each time level. Then, we derive some essential stability estimates and discuss the uniqueness of the approximations. In addition, we employ Nochetto mathematical framework to prove an optimal error bound in time for d= 1, 2 and 3 space dimensions. Finally, we present some numerical experiments to verify the obtained theoretical results.

Keywords: reaction diffusion system, finite element approximation, stability estimates, error bound

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Authors: Hassan Al Salman, Ahmed Al Ghafli

Abstract:

In this study we consider a nonlinear in time finite element approximation of a reaction diffusion system of lambda-omega type. We use a fixed point theorem to prove existence of the approximations. Then, we derive some essential stability estimates and discuss the uniqueness of the approximations. Also, we prove an optimal error bound in time for d=1, 2 and 3 space dimensions. Finally, we present some numerical experiments to verify the theoretical results.

Keywords: reaction diffusion system, finite element approximation, fixed point theorem, an optimal error bound

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Authors: Md. S. Ansari, S. S. Motsa

Abstract:

In this paper, an analysis has been made on the flow of non-Newtonian viscoelastic nanofluid over a linearly stretching sheet under the influence of uniform magnetic field. Heat transfer characteristics is analyzed taking into the effect of nonlinear radiation and non-uniform heat source/sink. Transport equations contain the simultaneous effects of Brownian motion and thermophoretic diffusion of nanoparticles. The relevant partial differential equations are non-dimensionalized and transformed into ordinary differential equations by using appropriate similarity transformations. The transformed, highly nonlinear, ordinary differential equations are solved by spectral local linearisation method. The numerical convergence, error and stability analysis of iteration schemes are presented. The effects of different controlling parameters, namely, radiation, space and temperature-dependent heat source/sink, Brownian motion, thermophoresis, viscoelastic, Lewis number and the magnetic force parameter on the flow field, heat transfer characteristics and nanoparticles concentration are examined. The present investigation has many industrial and engineering applications in the fields of coatings and suspensions, cooling of metallic plates, oils and grease, paper production, coal water or coal–oil slurries, heat exchangers’ technology, and materials’ processing and exploiting.

Keywords: magnetic field, nonlinear radiation, non-uniform heat source/sink, similar solution, spectral local linearisation method, Rosseland diffusion approximation

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Authors: Kankeyanathan Kannan

Abstract:

On approximation properties of group C* algebras is everywhere; it is powerful, important, backbone of countless breakthroughs. For a discrete group G, let A(G) denote its Fourier algebra, and let M₀A(G) denote the space of completely bounded Fourier multipliers on G. An approximate identity on G is a sequence (Φn) of finitely supported functions such that (Φn) uniformly converge to constant function 1 In this paper we prove that approximation property pass to free product.

Keywords: approximation property, weakly amenable, strong invariant approximation property, invariant approximation property

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Authors: Ali Safdari-Vaighani

Abstract:

Pricing financial contracts on several underlying assets received more and more interest as a demand for complex derivatives. The option pricing under asset price involving jump diffusion processes leads to the partial integral differential equation (PIDEs), which is an extension of the Black-Scholes PDE with a new integral term. The aim of this paper is to show how basket option prices in the jump diffusion models, mainly on the Merton model, can be computed using RBF based approximation methods. For a test problem, the RBF-PU method is applied for numerical solution of partial integral differential equation arising from the two-asset European vanilla put options. The numerical result shows the accuracy and efficiency of the presented method.

Keywords: basket option, jump diffusion, ‎radial basis function, RBF-PUM

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Authors: Salah Alrabeei, Mohammad Yousuf

Abstract:

The goal of option pricing theory is to help the investors to manage their money, enhance returns and control their financial future by theoretically valuing their options. However, most of the option pricing models have no analytical solution. Furthermore, not all the numerical methods are efficient to solve these models because they have nonsmoothing payoffs or discontinuous derivatives at the exercise price. In this paper, we solve the American option under jump diffusion models by using efficient time-dependent numerical methods. several techniques are integrated to reduced the overcome the computational complexity. Fast Fourier Transform (FFT) algorithm is used as a matrix-vector multiplication solver, which reduces the complexity from O(M2) into O(M logM). Partial fraction decomposition technique is applied to rational approximation schemes to overcome the complexity of inverting polynomial of matrices. The proposed method is easy to implement on serial or parallel versions. Numerical results are presented to prove the accuracy and efficiency of the proposed method.

Keywords: integral differential equations, jump–diffusion model, American options, rational approximation

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Authors: N. Guruprasad

Abstract:

This paper presents a modification of approximation method for transportation problems. The initial basic feasible solution can be computed using either Russel's or Vogel's approximation methods. Russell’s approximation method provides another excellent criterion that is still quick to implement on a computer (not manually) In most cases Russel's method yields a better initial solution, though it takes longer than Vogel's method (finding the next entering variable in Russel's method is in O(n1*n2), and in O(n1+n2) for Vogel's method). However, Russel's method normally has a lesser total running time because less pivots are required to reach the optimum for all but small problem sizes (n1+n2=~20). With this motivation behind we have incorporated a variation of the same – what we have proposed it has TMC (Total Modified Cost) to obtain fast and efficient solutions.

Keywords: computation, efficiency, modified cost, Russell’s approximation method, transportation, Vogel’s approximation method

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Authors: Ayan Chakraborty, BV. Rathish Kumar

Abstract:

Off-late many models in viscoelasticity, signal processing or anomalous diffusion equations are formulated in fractional calculus. Tempered fractional calculus is the generalization of fractional calculus and in the last few years several important partial differential equations occurring in the different field of science have been reconsidered in this term like diffusion wave equations, Schr$\ddot{o}$dinger equation and so on. In the present paper, a time-dependent tempered fractional diffusion equation of order $\gamma \in (0,1)$ with forcing function is considered. Existence, uniqueness, stability, and regularity of the solution has been proved. Crank-Nicolson discretization is used in the time direction. B spline finite element approximation is implemented. Generally, B-splines basis are useful for representing the geometry of a finite element model, interfacing a finite element analysis program. By utilizing this technique a priori space-time estimate in finite element analysis has been derived and we proved that the convergent order is $\mathcal{O}(h²+T²)$ where $h$ is the space step size and $T$ is the time. A couple of numerical examples have been presented to confirm the accuracy of theoretical results. Finally, we conclude that the studied method is useful for solving tempered fractional diffusion equations.

Keywords: B-spline finite element, error estimates, Gronwall's lemma, stability, tempered fractional

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Authors: Mansour Al Qubeissi, Sergei S. Sazhin, Cyril Crua, Morgan R. Heikal

Abstract:

This paper presents the application of the Discrete Component Model for heating and evaporation to multi-component biodiesel fuel droplets in direct injection internal combustion engines. This model takes into account the effects of temperature gradient, recirculation and species diffusion inside droplets. A distinctive feature of the model used in the analysis is that it is based on the analytical solutions to the temperature and species diffusion equations inside the droplets. Nineteen types of biodiesel fuels are considered. It is shown that a simplistic model, based on the approximation of biodiesel fuel by a single component or ignoring the diffusion of components of biodiesel fuel, leads to noticeable errors in predicted droplet evaporation time and time evolution of droplet surface temperature and radius.

Keywords: heat/mass transfer, biodiesel, multi-component fuel, droplet

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Authors: Alouaoui Redha, Ferhat Samira, Bouaziz Mohamed Najib

Abstract:

In this work, we examine the thermal radiation effect on heat and mass transfer in steady laminar boundary layer flow of an incompressible viscous micropolar fluid over a vertical plate, with the presence of a magnetic field. Rosseland approximation is applied to describe the radiative heat flux in the energy equation. The resulting similarity equations are solved numerically. Many results are obtained and representative set is displayed graphically to illustrate the influence of the various parameters on different profiles. The conclusion is drawn that the flow field, temperature, concentration and microrotation as well as the skin friction coefficient and the both local Nusselt and local Sherwood numbers are significantly influenced by Magnetic parameter, material parameter and thermal radiation parameter.

Keywords: MHD, micropolar fluid, thermal radiation, heat and mass transfer, boundary layer

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Authors: Chaghoub Soraya, Zhang Xiaoyan

Abstract:

This research presents the first constant approximation algorithm to the p-median network design problem with multiple cable types. This problem was addressed with a single cable type and there is a bifactor approximation algorithm for the problem. To the best of our knowledge, the algorithm proposed in this paper is the first constant approximation algorithm for the p-median network design with multiple cable types. The addressed problem is a combination of two well studied problems which are p-median problem and network design problem. The introduced algorithm is a random sampling approximation algorithm of constant factor which is conceived by using some random sampling techniques form the literature. It is based on a redistribution Lemma from the literature and a steiner tree problem as a subproblem. This algorithm is simple, and it relies on the notions of random sampling and probability. The proposed approach gives an approximation solution with one constant ratio without violating any of the constraints, in contrast to the one proposed in the literature. This paper provides a (21 + 2)-approximation algorithm for the p-median network design problem with multiple cable types using random sampling techniques.

Keywords: approximation algorithms, buy-at-bulk, combinatorial optimization, network design, p-median

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Authors: Valeria Bondarenko

Abstract:

In this paper, we propose two problems related to fractal Brownian motion. First problem is simultaneous estimation of two parameters, Hurst exponent and the volatility, that describe this random process. Numerical tests for the simulated fBm provided an efficient method. Second problem is approximation of the increments of the observed time series by a power function by increments from the fractional Brownian motion. Approximation and estimation are shown on the example of real data, daily deposit interest rates.

Keywords: fractional Brownian motion, Gausssian processes, approximation, time series, estimation of properties of the model

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Authors: Smita Sonker, Uaday Singh

Abstract:

Various investigators have determined the degree of approximation of functions belonging to the classes W(L r , ξ(t)), Lip(ξ(t), r), Lip(α, r), and Lipα using different summability methods with monotonocity conditions. Recently, Lal has determined the degree of approximation of the functions belonging to Lipα and W(L r , ξ(t)) classes by using Ces`aro-N¨orlund (C 1 .Np)- summability with non-increasing weights {pn}. In this paper, we shall determine the degree of approximation of 2π - periodic functions f belonging to the function classes Lipα and W(L r , ξ(t)) by C 1 .T - means of Fourier series of f. Our theorems generalize the results of Lal and we also improve these results in the light off. From our results, we also derive some corollaries.

Keywords: Lipschitz classes, product matrix operator, signals, trigonometric Fourier approximation

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Authors: N. Lebga, Kh. Bouamama, K. Kassali

Abstract:

We report first-principles calculation results on the structural and elastic properties of ZnS x Se1−x alloy for which we employed the virtual crystal approximation provided with the ABINIT program. The calculations done using density functional theory within the local density approximation and employing the virtual-crystal approximation, we made a comparative study between the numerical results obtained from ab-initio calculation using ABINIT or Wien2k within the Density Functional Theory framework with either Local Density Approximation or Generalized Gradient approximation and the pseudo-potential plane-wave method with the Hartwigzen Goedecker Hutter scheme potentials. It is found that the lattice parameter, the phase transition pressure, and the elastic constants (and their derivative with respect to the pressure) follow a quadratic law in x. The variation of the elastic constants is also numerically studied and the phase transformations are discussed in relation to the mechanical stability criteria.

Keywords: density functional theory, elastic properties, ZnS, ZnSe,

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Authors: Kamel Al-Khaled

Abstract:

Reaction-diffusion equations are commonly used in population biology to model the spread of biological species. In this paper, we propose a fractional reaction-diffusion equation, where the classical second derivative diffusion term is replaced by a fractional derivative of order less than two. Based on the symbolic computation system Mathematica, Adomian decomposition method, developed for fractional differential equations, is directly extended to derive explicit and numerical solutions of space fractional reaction-diffusion equations. The fractional derivative is described in the Caputo sense. Finally, the recent appearance of fractional reaction-diffusion equations as models in some fields such as cell biology, chemistry, physics, and finance, makes it necessary to apply the results reported here to some numerical examples.

Keywords: fractional partial differential equations, reaction-diffusion equations, adomian decomposition, biological species

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Authors: M. T. Ahn, J. H. Park, S. H. Park, S. H. Ha

Abstract:

Diffusion bonding has been continuously studied. Temperature and pressure are the most important factors to increase the strength between diffusion bonded interfaces. Diffusion bonding is an important factor affecting the bonding strength of the lined pipe. The increase of the diffusion bonding force results in a high formability clad pipe. However, in the case of drawing, it is difficult to obtain a high pressure between materials due to a relatively small reduction in cross-section, and it is difficult to prevent elongation or to tear of material in hot drawing even if the reduction in the section is increased. In this paper, to increase the diffusion bonding force, we derive optimal temperature and pressure to suppress material stretching and realize precise thickness precision.

Keywords: diffusion bonding, temperature, pressure, drawing speed

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Authors: Asma A. Parlin, K. Nakamura, N. Watanabe, T. Komai

Abstract:

Understanding the effective diffusion of benzene vapor in the soil-atmosphere interface is important as an intrusion of benzene into the atmosphere from the soil is largely driven by diffusion. To analyze the vertical one dimensional effective diffusion of benzene vapor in porous medium with high water content, diffusion experiments were conducted in soil columns using Andosol soil and Toyoura silica sand with different water content; for soil water content was from 0 to 30 wt.% and for sand it was from 0.06 to 10 wt.%. In soil, a linear relation was found between water content and effective diffusion coefficient while the effective diffusion coefficient didn’t change in the sand with increasing water. A numerical transport model following unsteady-state approaches based on Fick’s second law was used to match the required time for a steady state of the gas phase concentration profile of benzene to the experimentally measured concentration profile gas phase in the column. The result highlighted that both the water content and porosity might increase vertical diffusion of benzene vapor in soil.

Keywords: benzene vapor-phase, effective diffusion, subsurface soil medium, unsteady state

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Authors: Anusuya Ghosh, Vishnu Narayanan

Abstract:

The approximation of convex set by semidefinite representable set plays an important role in semidefinite programming, especially in modern convex optimization. To optimize a linear function over a convex set is a hard problem. But optimizing the linear function over the semidefinite representable set which approximates the convex set is easy to solve as there exists numerous efficient algorithms to solve semidefinite programming problems. So, our approximation technique is significant in optimization. We develop a technique to approximate any closed convex set, say K by compactly semidefinite representable set. Further we prove that there exists a sequence of compactly semidefinite representable sets which give tighter approximation of the closed convex set, K gradually. We discuss about the convergence of the sequence of compactly semidefinite representable sets to closed convex set K. The recession cone of K and the recession cone of the compactly semidefinite representable set are equal. So, we say that the sequence of compactly semidefinite representable sets converge strongly to the closed convex set. Thus, this approximation technique is very useful development in semidefinite programming.

Keywords: semidefinite programming, semidefinite representable set, compactly semidefinite representable set, approximation

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Authors: Tulin Coskun

Abstract:

We investigate the approximation of analytic functions of several variables in polydisc by the sequences of linear k-positive operators in Gadjiev sence. The approximation of analytic functions of complex variable by linear k-positive operators was tackled, and k-positive operators and formulated theorems of Korovkin's type for these operators in the space of analytic functions on the unit disc were introduced in the past. Recently, very general results on convergence of the sequences of linear k-positive operators on a simply connected bounded domain within the space of analytic functions were proved. In this presentation, we extend some of these results to the approximation of analytic functions of several complex variables by sequences of linear k-positive operators.

Keywords: analytic functions, approximation of analytic functions, Linear k-positive operators, Korovkin type theorems

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Authors: Ebenezer O. Ige, Funmilayo H. Oyelami, Joshua Olutayo-Irheren, Joseph T. Okunlola

Abstract:

Hyperthermia therapy is an adjuvant procedure during which perfused body tissues is subjected to elevated range of temperature in bid to achieve improved drug potency and efficacy of cancer treatment. While a selected class of hyperthermia techniques is shouldered on the thermal radiations derived from single-sourced electro-radiation measures, there are deliberations on conjugating dual radiation field sources in an attempt to improve the delivery of therapy procedure. This paper numerically explores the thermal effectiveness of combined infrared hyperemia having nanoparticle recirculation in the vicinity of imposed magnetic field on subcutaneous strata of a model lesion as ablation scheme. An elaborate Spectral relaxation method (SRM) was formulated to handle equation of coupled momentum and thermal equilibrium in the blood-perfused tissue domain of a spongy fibrous tissue. Thermal diffusion regimes in the presence of external magnetic field imposition were described leveraging on the renowned Roseland diffusion approximation to delineate the impact of radiative flux within the computational domain. The contribution of tissue sponginess was examined using mechanics of pore-scale porosity over a selected of clinical informed scenarios. Our observations showed for a substantial depth of spongy lesion, magnetic field architecture constitute the control regimes of hemodynamics in the blood-tissue interface while facilitating thermal transport across the depth of the model lesion. This parameter-indicator could be utilized to control the dispensing of hyperthermia treatment in intravenous perfused tissue.

Keywords: spectra relaxation scheme, thermal equilibrium, Roseland diffusion approximation, hyperthermia therapy

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Authors: Hare Krishna Nigam

Abstract:

In this paper, for the first time, (E,q)(C,2) product summability method is introduced and two quite new results on degree of approximation of the function f belonging to Lip (alpha,r)class and W(L(r), xi(t)) class by (E,q)(C,2) product means of Fourier series, has been obtained.

Keywords: Degree of approximation, (E, q)(C, 2) means, Fourier series, Lebesgue integral, Lip (alpha, r)class, W(L(r), xi(t))class of functions

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Authors: Kairat T. Mynbaev, Elena N. Lomakina

Abstract:

We consider a Hardy-type operator generated by a family of open subsets of a Hausdorff topological space. The family is indexed with non-negative real numbers and is totally ordered. For this operator, we obtain two-sided bounds of its norm, a compactness criterion, and bounds for its approximation numbers. Previously, bounds for its approximation numbers have been established only in the one-dimensional case, while we do not impose any restrictions on the dimension of the Hausdorff space. The bounds for the norm and conditions for compactness earlier have been found using different methods by G. Sinnamon and K. Mynbaev. Our approach is different in that we use domain partitions for all problems under consideration.

Keywords: approximation numbers, boundedness and compactness, multidimensional Hardy operator, Hausdorff topological space

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Authors: Kejal Khatri, Vishnu Narayan Mishra

Abstract:

We compute the degree of approximation of functions\tilde{f}\in H_w, a new Banach space using (T.E^1) summability means of conjugate Fourier series. In this paper, we extend the results of Singh and Mahajan which in turn generalizes the result of Lal and Yadav. Some corollaries have also been deduced from our main theorem and particular cases.

Keywords: conjugate Fourier series, degree of approximation, Hölder metric, matrix summability, product summability

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Authors: Shangerganesh Lingeshwaran

Abstract:

In this presentation, we have performed numerical simulations for a reaction-diffusion equation with various nonlinear density-dependent diffusion operators and proliferation functions. The mathematical model represented by parabolic partial differential equation is considered to study the invasion of gliomas (the most common type of brain tumors) and to describe the growth of cancer cells and response to their treatment. The unknown quantity of the given reaction-diffusion equation is the density of cancer cells and the mathematical model based on the proliferation and migration of glioma cells. A standard Galerkin finite element method is used to perform the numerical simulations of the given model. Finally, important observations on the each of nonlinear diffusion functions and proliferation functions are presented with the help of computational results.

Keywords: glioma invasion, nonlinear diffusion, reaction-diffusion, finite eleament method

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Authors: Ahmed Abdulrahman, Jalal Foroozesh

Abstract:

CO₂ based enhanced oil recovery (EOR) techniques have gained massive attention from major oil firms since they resolve the industry's two main concerns of CO₂ contribution to the greenhouse effect and the declined oil production. Carbonated water injection (CWI) is a promising EOR technique that promotes safe and economic CO₂ storage; moreover, it mitigates the pitfalls of CO₂ injection, which include low sweep efficiency, early CO₂ breakthrough, and the risk of CO₂ leakage in fractured formations. One of the main challenges that hinder the wide adoption of this EOR technique is the complexity of accurate modeling of the kinetics of CO₂ mass transfer. The mechanisms of CO₂ mass transfer during CWI include the slow and gradual cross-phase CO₂ diffusion from carbonated water (CW) to the oil phase and the CO₂ dispersion (within phase diffusion and mechanical mixing), which affects the oil physical properties and the spatial spreading of CO₂ inside the reservoir. A 2D non-equilibrium compositional simulator has been developed using a fully implicit finite difference approximation. The material balance term (k) was added to the governing equation to account for the slow cross-phase diffusion of CO₂ from CW to the oil within the gird cell. Also, longitudinal and transverse dispersion coefficients have been added to account for CO₂ spatial distribution inside the oil phase. The CO₂-oil diffusion coefficient was calculated using the Sigmund correlation, while a scale-dependent dispersivity was used to calculate CO₂ mechanical mixing. It was found that the CO₂-oil diffusion mechanism has a minor impact on oil recovery, but it tends to increase the amount of CO₂ stored inside the formation and slightly alters the residual oil properties. On the other hand, the mechanical mixing mechanism has a huge impact on CO₂ spatial spreading (accurate prediction of CO₂ production) and the noticeable change in oil physical properties tends to increase the recovery factor. A sensitivity analysis has been done to investigate the effect of formation heterogeneity (porosity, permeability) and injection rate, it was found that the formation heterogeneity tends to increase CO₂ dispersion coefficients, and a low injection rate should be implemented during CWI.

Keywords: CO₂ mass transfer, carbonated water injection, CO₂ dispersion, CO₂ diffusion, cross phase CO₂ diffusion, within phase CO2 diffusion, CO₂ mechanical mixing, non-equilibrium simulation

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Authors: Mei-Hsiu Chi, Jyh-Yang Wu, Sheng-Gwo Chen

Abstract:

The diffusion-reaction equations are important Partial Differential Equations in mathematical biology, material science, physics, and so on. However, finding efficient numerical methods for diffusion-reaction systems on curved surfaces is still an important and difficult problem. The purpose of this paper is to present a convergent geometric method for solving the reaction-diffusion equations on closed surfaces by an O(r)-LTL configuration method. The O(r)-LTL configuration method combining the local tangential lifting technique and configuration equations is an effective method to estimate differential quantities on curved surfaces. Since estimating the Laplace-Beltrami operator is an important task for solving the reaction-diffusion equations on surfaces, we use the local tangential lifting method and a generalized finite difference method to approximate the Laplace-Beltrami operators and we solve this reaction-diffusion system on closed surfaces. Our method is not only conceptually simple, but also easy to implement.

Keywords: closed surfaces, high-order approachs, numerical solutions, reaction-diffusion systems

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Authors: Mok-Tan Ahn, Joon-Hong Park, Yeon-Jong Jeong

Abstract:

Diffusion bonding has been continuously studied. Temperature and pressure are the most important factors to increase the strength between diffusion bonded interfaces. Diffusion bonding is an important factor affecting the bonding strength of the lined pipe. The increase of the diffusion bonding force results in a high formability clad pipe. However, in the case of drawing, it is difficult to obtain a high pressure between materials due to a relatively small reduction in cross-section, and it is difficult to prevent elongation or to tear of material in heat drawing even if the reduction in section is increased. In this paper, to increase the diffusion bonding force, we derive optimal temperature and pressure to suppress material stretching and realize precise thickness precision.

Keywords: drawing speed, FEM (Finite Element Method), diffusion bonding, temperature, heat drawing, lined pipe

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Authors: R. Whalley

Abstract:

The heat transfer modelling for a diffusion process will be considered. Difficulties in computing the time-distance dynamics of the representation will be addressed. Incomplete and irrational Laplace function will be identified as the computational issue. Alternative approaches to the response evaluation process will be provided. An illustration application problem will be presented. Graphical results confirming the theoretical procedures employed will be provided.

Keywords: heat, transfer, diffusion, modelling, computation

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