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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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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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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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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[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

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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,

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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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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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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.

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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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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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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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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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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

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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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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.

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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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