# Top PDF Solution of the two- dimensional heat equation for a rectangular plate

### Solution of the two- dimensional heat equation for a rectangular plate

Abstract: Laplace equation is a fundamental equation of applied mathematics. Important phenomena in engineering and physics, such as steady-state temperature distribution, electrostatic potential and fluid flow, are modeled by means of this equation. The Laplace equation which satisfies boundary values is known as the Dirichlet problem. The solutions to the Dirichlet problem form one of the most celebrated topics in the area of applied mathematics. In this study, a novel method is presented for the solution of two-dimensional heat equation for a rectangular plate. In this alternative method, the solution function of the problem is based on the Green function, and therefore on elliptic functions.
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### On the General Solution for the Two-Dimensional Electrical Impedance Equation in Terms of Taylor Series in Formal Powers

The two-dimensional case of Calderon problem is specially interesting for medical image reconstruction, and it kept the attention of many researchers from its very appearing. Beside the purely numerical approaches, a good alternative for solving the Electrical Impedance Tomography problem is to use a wide class of analytic solutions for (1), posing different conductivity functions, and comparing such solutions valued in the boundary points, with the collected data until the difference can be considered minimum. Nevertheless, the mathematical complexity for solving analytically (1) represented such a challenge, that many experts considered impossible to obtain its general solution in analytic form [5], even for the simplest cases of (not including the constant case, of course).
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### Analytical representation of the solution of the space kinetic diffusion equation in a one-dimensional and homogeneous domain

In this work we determine a new solution free of stiffness and in analytical representation in one- dimensional geometry and a homogeneous domain. In summary, the methodology consists of ex- panding the scalar neutron flux and the concentration of delayed neutron precursors in Taylor series in the spatial variable (applying the methodology discussed in (CEOLIN, 2014), where the temporal dependence is incorporated in the coefficients of that series, that allows to decompose the original problem into a recursive system of time-dependent ordinary differential equations. We avert the stiff- ness character using the idea of (SILVA et al., 2014) (applied in the solution of the problem of neutron point kinetics) and thus obtaining the solution of the problem. The idea consists in splitting the co- efficient matrix into two, one constant diagonal matrix and the second one with the remaining time dependent and off-diagonal terms. Moreover, the equation system is reorganized such that the terms containing the second matrix are assigned as source terms. The homogeneous equation system has a well known solution, since the matrix is diagonal and constant, and plays the role of the recursion initialization of the decomposition method (ADOMIAN, 1988; ADOMIAN and RACH, 1996; ADOMIAN, 1994; PETERSEN, 2011; SILVA et al., 2014). The recursion scheme is set up in a fashion where the solutions of the previous recursion steps determine the source terms of all subse- quent steps. A second feature of the method is the choice satisfying the initial and boundary conditions by the recursion initialization, so that from the first recursion step onward initial and boundary con- ditions are homogeneous. The fact that the time evolution in the solution is calculated recursively for in principle all times does not impose any restrictions such as convergence limitations that are typi- cally present in progressive time step approaches, used by the methods cited above and elsewhere.
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### An Optimized Numerical Method for Solving the Two-Dimensional Impedance Equation

The relevance of efficiently solving the forward problem for (1), if we are to solve the Electrical Impedance Tomog- raphy problem (also called inverse problem), was widely exposed in a variety of works, among which [12] is one of the most important. In this sense, the results posed in [8], and subsequently rediscovered in [1], are indeed very significant, because they allowed to sift out the rink for approaching the general solution of the Impedance Equation in the plane.

### Mathematical Model for Fluid Flow and Heat Transfer Processes in Plate Exchanger

values of pressure and temperature distribution, for estimated values of velocity in x and y direction, in area of definition, under known boundary conditions. For system equation solution, on that way of problem formulation, two methods are known. The direct methods, where Gauss method of elimination belongs and Cramer rule are characterized with large number of arithmetical operations and a lot of time for calculation for getting results. On the other hand, iterative methods are more simplified for programing, the calculation takes short time with satisfactory accuracy, [2], [3].
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### A Comparative Study of Numerical Techniques for 2d Transient Heat Conduction Equation Using Finite Element Method

Nihad Dukhan [3] presented a one-dimensional heat transfer model for open-cell metal foam, combining the conduction in the ligaments and the convection to the coolant in the pores. Comini and Guide [see 4] proposed a general applicable approach using non-linear physical properties and boundary conditions for transient heat conduction problem using triangular elements for space discretization and using Crank-Nicholson algorithm for each time step. [5] Solved the two dimensional parabolic problem by considering heat conduction in a slab. A space-time finite element has been applied using linear hexahedral elements in space-time domain. Sutradhar [6] found transient temperature distribution for homogeneous and non-homogeneous materials using Laplace transform Galerkin boundary element method. [7] Studied Finite Element Weighted Residual technique for non-linear two-dimensional heat problems using rectangular prism. In the present work a two-dimensional transient heat flow has been considered for solids. A mathematical model has been constructed so that temperature variation can be studied everywhere inside the domain. Solution is started with reformulation of the given differential equation as an equivalent variational problem. The special feature of the finite element method is that the functions are chosen to be piecewise polynomials. Triangular and rectangular finite elements are used. Comparative study has been made taking different combinations of meshes and the appropriate space-time FEM techniques.
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### On the Numerical Solution of Three-Dimensional Diffusion Equation with an Integral Condition

[9] A. Akram and M.A. Pasha, Numerical Method for the Heat Equation with a Non Local Boundary Condition. International Journal of Information and systems Sciences, Vol 1, Number 2 (2005) 162-171. [10] A. B. Gumel, W. T. Ang and F. H. Twizell.”Efficient Parallel

### A solution of two-dimensional magnetohydrodynamic flow using the finite volume method

The difficulty is that in the previous equations there are two unknown: the pressure and the velocity. The elimination of pressure from the equations leads to a vorticity-stream function which is one of the most popular methods for solving the 2-D incompressible Navier-Stokes equation

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### Solution of two-dimensional Fredholm integral equation via RBF-triangular method

In recent years, some numerical methods have been proposed to estimate the solution of one-dimensional and two-dimensional integral equations such as [1, 2, 7, 8]. RBFs played an important role in approximation theory to introduce a new basis in numerical solution of integral equations [3, 5, 9, 10]. In this work, the Gaussian radial basis functions (RBFs) is applied to solve the two-dimensional Fredholm integral equation of the second kind as follows,

### Numerical Solution of Heat Equation with

ABSTRACT. In this work we study the numerical solution of one-dimensional heat diffusion equation subject to Robin boundary conditions multiplied with a small parameter epsilon greater than zero. The numerical evidences tell us that the numerical solution of the differential equation with Robin boundary condition are very close in certain sense of the analytic solution of the problem with homogeneous Dirichlet boundary conditions when ε tends to zero.

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### Scattering of Fexural Gravity Waves by a Two-Dimensional Thin Plate

important. Mathematically, the boundary value problem (BVP) related to study of water waves in ocean with ice-cover, involves ﬁfth order derivative of the potential function in the boundary condition on ice cover whereas the governing partial differential equation is of second order. The literature concerning the study of ocean wave interaction in ocean with ice-cover in the presence of a body submerged beneath the ice-cover ﬂoating in a deep water is rather limited, although the study of ocean wave interaction with structures present in the ocean with free surface under linearised theory has been a subject of interest since early twentieth century. A number of researchers contributed signiﬁcantly to this topic, although the closed form solution to these problems are available only when the structure is in form of a thin rigid vertical plate and that too for the two dimensional motion in water. Diffraction problems involving nearly vertical barriers are more general than vertical barrier. One such problem of water waves scattering by a nearly vertical plate partially immersed in deep water was considered by Shaw (1985). He used a perturbation analysis that involved solution of singular integral equation. Later Mandal and Chakrabarti (1989) and Mandal and Kundu (1990) considered the problems of water waves scattering by a nearly vertical barrier and utilized a perturbation analysis different from Shaw (1985) to handle the problems. The problem of water wave diffraction by a symmetric two dimensional thin slender was plate mentioned brieﬂy by Shaw (1985) although the ﬁrst order correction to reﬂection and transmission coefﬁcients are not given there explicitly. Later Kundu (1997), Kundu and Saha (1998) considered the problem of water wave scattering by a thin two dimensional slender body either partially immersed or completely submerged or submerged in deep water. They used the perturbation technique described in Mandal and
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### Finite Element Analysis of the Crack Propagation for Solid Materials

Abstract: Problem statement: The use of fracture mechanics techniques in the assessment of performance and reliability of structure is on increase and the prediction of crack propagation in structure play important part. The finite element method is widely used for the evaluation of SIF for various types of crack configurations. Source code program of two-dimensional finite element model had been developed, to demonstrate the capability and its limitations, in predicting the crack propagation trajectory and the SIF values under linear elastic fracture analysis. Approach: Two different geometries were used on this finite element model in order, to analyze the reliability of this program on the crack propagation in linear and nonlinear elastic fracture mechanics. These geometries were namely; a rectangular plate with crack emanating from square-hole and Double Edge Notched Plate (DENT). Where, both geometries are in tensile loading and under mode I conditions. In addition, the source code program of this model was written by FORTRAN language. Therefore, a Displacement Extrapolation Technique (DET) was employed particularly, to predict the crack propagations directions and to, calculate the Stress Intensity Factors (SIFs). Furthermore, the mesh for the finite elements was the unstructured type; generated using the advancing front method. And, the global h-type adaptive mesh was adopted based on the norm stress error estimator. While, the quarter- point singular elements were uniformly generated around the crack tip in the form of a rosette. Moreover, make a comparison between this current study with other relevant and published research study. Results: The application of the source code program of 2-D finite element model showed a significant result on linear elastic fracture mechanics. Based on the findings of the two different geometries from the current study, the result showed a good agreement. And, it seems like very close compare to the other published results. Conclusion: A developed a source program of finite element model showed that is capable of demonstrating the SIF evaluation and the crack path direction satisfactorily. Therefore, the numerical finite element analysis with displacement extrapolation method, had been successfully employed for linear-elastic fracture mechanics problems.
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### A Review on Heat Transfer Improvent of Plate Heat Exchanger

that even at moderate velocities plate heat exchanger can achieve high heat transfer coefficient, low fouling factor etc. Nusselt Number is found to be greatly depending upon the Reynolds Number and it increases with the increase in Reynolds Number. At the different possible conditions various correlations have been proposed for Nusselt Number, Reynolds Number, Prandtl Number, heat transfer coefficient, friction factors etc. Dimensionless correlations have also been proposed for the plate heat exchanger. Models have been developed for the study of compact heat exchanger with multiple passes and multiple rows for the development of better generalized equations.
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### The Economics of Sub-optimal Policies for Traffic Congestion

Closer to this paper is the work of Gallego, Montero, and Salas [2013]. These authors analyze two policies, in Mexico City and Santiago (Chile), aimed at reducing congestion and pollution. They find the policies that impose driving restrictions may lead to a higher number of cars on the city. Our results point on the same direction. Batarce and Ivaldi [2014] estimate the demand for transportation mode taking into account traffic congestion in an equilibrium setup. In their work, traffic congestion is the equilibrium of a game with a continuum of drivers. De Borger and Proost [2012] theoretically analyze the political economy aspects of congestion pricing. Their results corroborate the emprical observation that road pricing is politically difficult to implement. We analyze the same problem empirically.
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### Study of the enthalpy-entropy mechanism from water sorption of orange seeds (C. sinensis cv. Brazilian) for the use of agro-industrial residues as a possible source of vegetable oil production

Orange (C. sinensis), the most important of the Citrus fruits, is a tree growing to some 15 m in height. It perhaps originated in southern China as a hybrid between C. maxima and C. reticulata and was taken to Europe in the iteenth century. Sweet orange is grown throughout the subtropics and tropics, but Brazil and the United States of America produce the greatest quantities of this fruit. In both countries, the bulk of production is used to manufacture orange juice (VAUGHAN; GEISSLER, 2009). he fruit is a hesperidium, carpels, or segments illed with juicy arils and seeds. Seeds are white, show polyembryony, and vary in size and number in diferent species. Chemically, sweet oranges contain 6-9% of total sugars and 44-79 mg/100 g of vitamin C. In addition to being widely consumed as a fresh fruit, its juice is also a good source of sugars, vitamin C, and potassium (DOIJODE, 2001; UNIVERSIDADE..., 2006).
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### Field Cancerisation of the Upper Aerodigestive Tract: Screening for Second Primary Cancers of the Oesophagus in Cancer Survivors

multiple synchronous and/or metachronous cancers of the oesophagus, lungs, and head and neck region (i.e. oral cavity, oropharynx, hypopharynx, or larynx). 90% of the tumours in head and neck are squamous cell carcinomas, and at least 75% of them are attributable to the combination of tobacco and alcohol consumption. The odds ratio of OSCC may be as high as 50.1 for those who are both heavy smokers and heavy drinkers in comparison to people who neither drink nor smoke. 13 It has been estimated

### On the solution of non-linear diffusion equation

giving convergence of order m + 2. It is also called the Shamanskii method [14]. Optimal choices of m are problem dependent and affected from the computational cost ratio be- tween forming of the Jacobian matrix and of the residual vector. If the cost of updating the tangent matrix is high, the Shamanskii method is worthwhile. Numerical experiments show that the number of simplified Newton steps should be variable and usually increasing along the iteration number, e.g. like m = i where i is the number of a corrector iteration. However, m should have some upper limit for practical purposes. In this study m is limited to three.
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### New multi-soliton solutions for generalized Burgers-Huxley equation

This equation can be solved by various analytical methods, such as the variational iteration method [2], the homotopy perturbation method [3-5], and the exp-function method [6, 7]. A complete review on various analytical method is available in [8, 9]. In this paper the double exp-function method [10] is adopted to elucidate the different velocities and different frequencies in the travelling wave.

### Distribution of temperature in of visco-elastic plate with a washer imbedded under the effect of vibration

Distribution of temperature in of visco-elastic plate with a washer imbedded under the effect of vibration The distribution of temperature in an infinite plate of visco-elastic material with a circular hole,into which is embedded visko-elastic circular disc from another viaco-elastic material is examined. Applied load is a tensile force acting at infinity in the direction of the ox axis, which varies harmonically with constant amplitude. The case of omnidirectional tension of the plate is considered as well.
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