Abstract: Laplace equation is a fundamental equationof 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 ofthe most celebrated topics in the area of applied mathematics. In this study, a novel method is presented for thesolutionoftwo-dimensionalheatequation for a rectangularplate. In this alternative method, thesolution function ofthe problem is based on the Green function, and therefore on elliptic functions.
Thetwo-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 , even for the simplest cases of (not including the constant case, of course).
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 thesolutionofthe problem of neutron point kinetics) and thus obtaining thesolutionofthe 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, theequation 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 ofthe recursion initialization ofthe 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 ofthe previous recursion steps determine the source terms of all subse- quent steps. A second feature ofthe 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 thesolution 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.
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  is one ofthe most important. In this sense, the results posed in , and subsequently rediscovered in , are indeed very significant, because they allowed to sift out the rink for approaching the general solutionofthe Impedance Equation in the plane.
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 equationsolution, 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, , .
Nihad Dukhan  presented a one-dimensionalheat 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.  Solved thetwodimensional 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  found transient temperature distribution for homogeneous and non-homogeneous materials using Laplace transform Galerkin boundary element method.  Studied Finite Element Weighted Residual technique for non-linear two-dimensionalheat 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 ofthe given differential equation as an equivalent variational problem. The special feature ofthe 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.
 A. Akram and M.A. Pasha, Numerical Method for theHeatEquation with a Non Local Boundary Condition. International Journal of Information and systems Sciences, Vol 1, Number 2 (2005) 162-171.  A. B. Gumel, W. T. Ang and F. H. Twizell.”Efficient Parallel
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 ofthe most popular methods for solving the 2-D incompressible Navier-Stokes equation
In recent years, some numerical methods have been proposed to estimate thesolutionof 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 solutionof integral equations [3, 5, 9, 10]. In this work, the Gaussian radial basis functions (RBFs) is applied to solve thetwo-dimensional Fredholm integral equationofthe second kind as follows,
ABSTRACT. In this work we study the numerical solutionof one-dimensionalheat diffusion equation subject to Robin boundary conditions multiplied with a small parameter epsilon greater than zero. The numerical evidences tell us that the numerical solutionofthe differential equation with Robin boundary condition are very close in certain sense ofthe analytic solutionofthe problem with homogeneous Dirichlet boundary conditions when ε tends to zero.
important. Mathematically, the boundary value problem (BVP) related to study of water waves in ocean with ice-cover, involves ﬁfth order derivative ofthe 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 thetwodimensional 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 solutionof 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 twodimensional 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 twodimensional slender body either partially immersed or completely submerged or submerged in deep water. They used the perturbation technique described in Mandal and
In the present study, by considering strain gradient theory, buckling analysis of thin rectangular functionally graded micro-plates was surveyed. Using variational approach and principle of mini- mum total potential energy, higher order governing equations were determined which contain the microstructure parameters. It was assumed micro-plate is made of functionally graded material with power law distribution of material properties through the thickness. Finally, the stability equation was solved analytically for a simply supported micro-plate and critical buckling loads were ob- tained. It was concluded that increasing the index of FGM decrease the non-dimensional critical buckling load. Also, increasing the microstructure parameter decreases the buckling load. It was inferred that load carrying capacity is greatly depends on the loading conditions. Accordingly, pres- ence of tensile load increases the load capacity. In addition, buckling may occur in higher modes, where the mode is affected by the aspect ratio or loading conditions.
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 oftwo-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 rectangularplate 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 ofthe source code program of 2-D finite element model showed a significant result on linear elastic fracture mechanics. Based on the findings ofthetwo 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.
that even at moderate velocities plateheat 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 theplateheat 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.
Closer to this paper is the work of Gallego, Montero, and Salas . 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  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  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.
Orange (C. sinensis), the most important ofthe 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).
multiple synchronous and/or metachronous cancers ofthe oesophagus, lungs, and head and neck region (i.e. oral cavity, oropharynx, hypopharynx, or larynx). 90% ofthe 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
giving convergence of order m + 2. It is also called the Shamanskii method . Optimal choices of m are problem dependent and affected from the computational cost ratio be- tween forming ofthe Jacobian matrix and ofthe 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.
This equation can be solved by various analytical methods, such as the variational iteration method , 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  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 The distribution of temperature in an infinite plateof 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 ofthe ox axis, which varies harmonically with constant amplitude. The case of omnidirectional tension oftheplate is considered as well.