Simultaneously, the good numerical approximations on Burgers’ equation also grew as time goes. Examples like Nyuyen and Reynen (1984) presented a space-time **finite** element approach to Burgers’ equation, Kakuda and Tosaka (1990) used generalized boundary element **method**, Bar-Yoseph et al. (1995) used and discussed the space time spectral element **method** on Burgers’s equation solution, Zhu et al. (2009) applied a cubic B-spline quasi interpolation to Burgers equation, Siraj et al.(2012) researched the numerical solution o f Burgers’ equations using meshless **Method** o f Lines and many more.

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Salinity refers to the amount of salt in rivers, where the salt can be in many diﬀerent forms. There are two main methods of deﬁning the concentration of salt in water such as the total dissolved solid measurement (TDS) and the electrical conductivity measurement (EC). The salinity is measured by evaporating water to dryness and weighing the solid residue. The electrical conductivity measurement is measured by passing an electric current through the water and measuring how readily the current ﬂows. The total amount of salt in the water can aﬀect the taste of water. The World Health Organization’s guideline on water palatability is that water with a salinity level of less than about 0.50–0.60 g/L is generally considered to be of a standard level. The drinking-water becomes signiﬁcantly and increasingly unpalatable at salinity levels greater than about 1.0 g/L. In this research, a one-dimensional mathematical model of salinity measurement in a river is proposed. A modiﬁed model of salinity control in a river with a barrage dam is also introduced. An unconditionally stable **explicit** ﬁnite diﬀerence technique is used to approximate the salinity level under several

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compared with chatterjee’s solution but it did not display good agreement, it may be due to limitations of solution in respect of time step (5 sec) and use of constant friction factor value in Chatterjee’s solution. Das (1997) developed numerical methods of solution for surge height analysis and compared with laboratory data. Das et al. (2005) also worked on the numerical solution using Barr’s (1981) resistance equation. In this study the numerical **method** developed is the “**Explicit** **finite** **difference** **method**”. The solution so developed takes care of friction by using three recent resistance equations along with Barr’s resistance equation. Result so obtained are compared with the Modified Jacobsen **method** and existing experimental work. A review of the previously reported experimental data was undertaken. In Experimental work of Wood (1976), the length of pipe was only 36 feet with diameter 1.025 inches. His data presented were not of kind as required to assess numerical solutions. Simpson and Wylie (1991) also conducted experiments in a 36 m long pipe having 19.05 mm diameter. Pressure pulses are presented graphically for short durations, thus this type of data was not considered. Martin (1983) also conducted experiment work on the column separations situations. AIT, Bangkok (1969) has provided experimental data from an experimental setup of a large constant head reservoir with a 2 inch diameter pipe and 4.5 inch diameter clear plastic surge tank. The length of the pipe was 28.76 feet and time interval was 1 to 28 seconds. Here, as the pipe length and time interval were very small, damping due to friction may not occur. Rao, P. V. et al. (1993) also studied on this work with laboratory set up. In this study laboratory data were taken from Das M. M. and Das Saikia M. (2016) for analysis. The experiment was conducted in Hydraulics laboratory of Assam Engineering College for time period up to 300 seconds.

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The **explicit** and hybrid methods were found to be more efficient than the fully im- plicit and Crank-Nicolson methods for all N , the number of spatial steps, over both a small and large second spatial derivative coefficient, K , due to the nonlinearity in the neural model. The **explicit** **finite** **difference** **method** has the advantage of simplicity of programming and for small K is very efficient for all N . For larger K, the hybrid **method** is the most efficient **finite** **difference** **method** once N passes a threshold. Both the fully implicit and the Crank-Nicolson methods are much less efficient due to the evaluation of the nonlinear term using Newton’s **method**. We then investigated using a Fourier series approximation to the neural model. It was found that the nonstiff ODE solvers are the most effective in solving the system for small K. As K increases and the stiffness of the system grows, ODE solvers for stiff systems become the most effective. For both small and large K, if a highly ac- curate approximation is required (a large number of Fourier coefficients), multistep solvers are the most efficient.

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Noye and Tan obtained the numerical solutions of the ADE by the third-order semi **explicit** **finite** **difference** **method** [31]. Various two-level **explicit** and implicit **finite** **difference** methods covering the upwind **explicit**, the Lax-Wendroff, the modified Siemieniuch-Gladwell and the fourth-order **method** have been compared with each other on the numerical solutions of model problems including the ADE. Karahan solved various initial boundary value problems for the ADE by the use of implicit, third-order upwind and **explicit** **finite** **difference** methods [19–21]. Guraslan et al. developed a sixth-order compact **finite** **difference** **method** (CD6) combined with the fourth order Runge-Kutta **method** for numerical solution of three dynamic model problems [11].

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The aim of this paper is to discretize the ice thickness equation of the model using the **explicit** **finite** **difference** **method** and performed the numerical calculation by using CUDA on GPU platform for parallel computing. In this era, the utilization of GPU is not only for powerful rendering the graphics, but also can be used for general purpose of non-graphical computing. GPU is outstanding in applying for both graphics and computation purpose. CUDA is the parallel programming with the use of GPU. CUDA has been introduced in 2006 as the first general computing that harnessing general purpose computing on graphics processing unit (GPGPU). This programming has been used by many researchers to solve their problem with the large scale and complex computational in a more efficient way. CUDA is flexible to utilize the resources of GPU due to a unified programming model. The high data-parallelism of CUDA enables to speedup the computation [4]. Thus, this study will be implemented and run the ice thickness equation on two computer models which are C programming on the Central Processing Unit (CPU) and CUDA embedded with C language on GPU. GPU computation platform is chosen to reduce the

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In chapter four, we discuss the implementation of one dimensional groundwater flow modeling for **explicit**, implicit and Crank-Nicolson **method**. The problem we solve by using MATLAB software. Then, the results obtain are compared to each **method** and displayed in tables.

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Abstract—Models for multiphase-multicomponent flow in porous media are described in systems of PDEs. Solving them using **finite** **difference** discretization **method** can be of three ways; **explicit**, semi-implicit, and fully implicit. In this study, we present the implementation of an adaptive Newton-Raphson **method** in the context of IMPECS (Implicit Pressures **Explicit** Concentrations and Saturations) **method** of solution.

The research on FSI in fluid-filled pipes started many years ago. In the numerical researches, solutions based on the **Method** of Characteristics (MOC), the **Finite** Element **Method** (FEM), the **Finite** Volume **Method** (FVM) or a combination of these, is predominant. Gale and Tiselj found Godunov’s **method** as a very promising numerical **method** for simulations of the FSI problems [5]. Lavooij and Tijsseling presented two different procedures for computing FSI effects: full MOC uses MOC for both hydraulic and structural equations and in MOC–FEM the hydraulic equations are solved by the MOC and the structural equations by the FEM [6]. Using the MOC–FEM approach, Ahmadi and Keramat studied various types of junction coupling but MOC suffers from restrictions on linearity of equations and space-time mesh sizing [7].

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The numerical methods that have been applied to advection-diffusion equation include **finite** **difference** me- thods, Galerkin methods, spectral methods, wavelet based **finite** elements and several others. The AGE **method** is an iterative **method** employing the fractional splitting strategy which is applied alternately at each interme- diate step on tridiagonal system of **difference** schemes. Its rate of convergence is governed by the acceleration parameter r . The AGE iterative **method** is applied to a variety of problems involving parabolic and hyperbolic partial differential equations (see [3]-[6]). In [7], Sahimi and Evans reformulated the AGE **method** to solve the Navier-Stokes equations in the stream function-vorticity form. In this paper we apply the AGE iterative **method** to the one-dimensional advection diffusion equation. We put forward a new iterative **method** by numerical dif- ferentiation and also extend the original **method**. The AGE **method** is shown to be extremely powerful and flexi- ble and affords its users many advantages. Computational results are obtained to demonstrate the applicability of the **method** on some problems with known solutions.

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experimental and three different numerical models of the **method** of characteristics model (MOC), the axisymmetrical model and the implicit scheme model. The results of different computer models agree well with the experimental data [2]. Afshar and Rohani applied an implicit MOC to a problem of transient flow caused by the failure of a pump with and without a check valve and compared the results with those of the **explicit** MOC. Results showed that the implicit MOC can be used for any combination of devices to accurately predict the variations of head and flow in the pipeline system [3]. Sabbagh-Yazdi et al. applied a second-order **explicit** Godunov- type scheme to water hammer problems. The minimum and maximum of the computed pressure waves were in close agreement with the analytical solution and laboratory data [4]. However, the **method** still fails in the precise prediction of discontinuities. Zhao and Ghidaoui applied first- and second-order Godunov-type schemes for water hammer problems. Numerical tests showed that the first-order Godunov gives the same results to the MOC with space-line interpolation [5]. Chaudhry and Hussaini solved water hammer equations by three **explicit** **finite**- **difference** schemes (MacCormack’s **method**, Lambda scheme and Gabutti scheme). Their study revealed that for the same accuracy, second-order schemes required fewer computational nodes and less computer time as compared to those required by the first-order MOC [6]. Tijsseling and Bergant proposed a **method** based on the MOC, but a numerical grid is not required. The water hammer equations without friction have been solved exactly for the time-dependent boundary and constant (steady state) initial conditions with this **method**. Their **method** was the simplicity of the algorithm (recursion) and the fast and accurate (exact) calculation of transient events but calculation time strongly increased the events of longer duration [7]. Hou et al. [8] simulated water hammer with the corrective smoothed particle **method** (CSPM). The CSPM results are in good agreement with conventional MOC solutions.

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In recent years, there have been many research achievements on the numeri- cal algorithm of fractional diffusion equations. Xu Chuanju et al. applied the spectral **method** directly to the solution of time fractional derivatives, and proved the convergence by providing a priori error estimate [9]. Liu Fawang et al. proposed the implicit RBF **method** for solving the time fractional diffusion equation. Since no grid meshing is consistent with the definition of fractional derivatives, it has been regarded as a promising research direction [10]. However in the existing numerical algorithms, the **finite** **difference** **method** is still domi- nant. A class of unconditionally stable and convergent implicit **difference** ap- proximate **method** for fractional diffusion equation was constructed by Zhuang Pinghui et al. , which was second-order in space and 2 − α order in time [11]. Yuste established the forward Euler **difference** scheme and put forward the weighted average **finite** **difference** scheme by G-L (Grunwald-Letnikov) ap- proximation **method** for the time fractional sub-diffusion model, as well as proved the stability of the scheme [12]. Pu Hai et al. examined a C-N (Crank-Nicolson) **difference** **method** to solve a class of sub-diffusion equations with variable coefficients [13]. The approach was proved unconditionally stable by the energy **method** and convergent to temporally min 2 { − α 2,1 + α } order and spatially second order. Gao Guanghua et al. derived a compact **finite** differ- ence scheme for the sub-diffusion equation, which is fourth-order accuracy ap- proximation for the space derivative [14].

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Figure 8 illustrates von Mises stress history of different friction coefficients for Super CMV and Ti-6Al-4V materials. The trend of the graph shows that Von Mises stress is increasing simultaneously for all the friction coefficients during loading according to the material. The maximum stress is recorded during scratching for all the friction cases. Higher friction coefficient displays higher von Mises stress while lower stress is noted for friction coefficient of 0.3. Meanwhile, only an insignificant **difference** is noted between 0.6 and 0.9 respectively. The von Mises stress acting on Super CMV is higher than Ti-6Al-4V. The unloading process causes gradual drop in stress, where residual stress can be noticed.

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As the generalized **finite** **difference** approximation must be obtained at each point the approximations of the partial derivatives obtained by a local approximation (MLS) are much more accurate that those obtained by a global approximation (LS), see for example [7]. The MLS-GFDM is a truly mesh-free **method** using only points to determine the nodal supports. The main contribution of this paper is the application of an **explicit** scheme based in the MLS- GFDM to the case of modelling monodomain electrical conductivity problems using operator splitting including the case of anisotropic real tissues. The remainder of this paper is organized as follows. In Section 2 the monodomain model of cardiac tissue is introduced. In Section 3 we present how the monodomain model may be solved with an operator splitting **method**. In Section 4 the **explicit** GFDM is developed. Numerical experiments intended to validate the solution algorithm are presented in Section 5. Finally, in Section 6 some conclusions that can be drawn from the paper about the proposed **method**.

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Overall, what stands out from these examples is that the **finite** element **method** has better approximations for the first problem compared to **finite** **difference** **method** for all the step-sizes that we have. But in the second problem we have a small **difference** in the results with better accuracy for the **finite** **difference** **method** and in the third problem the **finite** element **method** has bigger relevant errors than the **difference** **method**. More specifically, in example 1 according to the tables we have better approximations for the **finite** element **method** in both of step sizes and the mesh size parameters to specific points but the graphs of the percentage of relevant errors suggest that the second order central **finite** **difference** scheme produce better approximations generally. On the other hand, in the other two examples according to the tables and the graphs of errors in the second problem we have a small **difference** between these methods and in the third problem we have better approximations of the second order central **difference** scheme almost in all points of the domain. Conclusively, we can notice that in the third example for both of these methods the results which we obtained are almost identical when different step sizes are applied in CFDM and mesh size parameters in FEM.

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If the interpolation points are chosen radially from the center of the interpolation point X until enough interior grid points (6 ∼ 9 for second order schemes, 10 ∼ 15 for third order scheme) are included, the **method** works fine if the curvature at X is not too large. However, if the curvature is large and there are fewer grid points in a thin tube of the normal direction compared with that of the tangential direction, then we may either need a large circle to include more interior points in the tube, or we may not have a good accuracy for the interpolated normal derivative. Either case will affect the efficiency of the numerical algorithm.

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The equation (8) is the original equation which we are solving. By using this equation we can formulate the nodal equation for finding the temperatures at the nodes where the data is unknown. Once the equation is formulated it can be convert into a matrix of form AT=X. This matrix can be solved by various methods like Gauss elimination **method**, Gauss-Seidel iterations, Thomas algorithm, Successive over relaxation **method** and Cramer’s **method** of solving the matrix, etc.

In El’sgol’ts [10], similar boundary value problems with solutions that show quick oscillations are considered. Kadalbajoo and Sharma [2],[3], Kadalbajoo and Ramesh [4] and Kadalbajoo and Kumar [5],[6] started a broad numerical work for solving singularly perturbed delay differential equations based on **finite** **difference** scheme, fitted mesh and B-spline **method**, piecewise uniform mesh. Lange and Miura [11],[12] gave asymptotic approaches in the study of class of boundary value problems for linear second order differential **difference** equations in which the highest order derivative is multiplied by small parameter. They have also additionally talked about the impact of small shifts on the oscillatory solution of the problem.Duressa et al. and Sirisha L et al. [13],[14] exhibited numerical methods to approximate the solution of boundary value problems for SPDDEs with delay as well as advance.

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We have derived the discretization of the chemotaxis-hypotax- is system (1) using a GFD scheme. The discrete solution obtained inherits the complicated dynamical behavior of the analytical solu- tion. The Generalized **Finite** **Difference** **Method** solves this strongly coupled highly nonlinear parabolic-elliptic system efficiently and with high accuracy.

Another technique which is discussed in this study is Differential Quadrature **Method** (DQM). As stated by (C. Shu, 2000), DQM is an extension of FDM for the highest order of **finite** **difference** scheme. This **method** represents by sum up all the derivatives of the function at any grid points, and then the equation transforms to a system of ordinary differential equations (ODEs) or a set of algebraic equations (R.C. Mittal and Ram Jiwari, 2009). The system of ordinary differential equations is then solved by numerical methods such as the implicit Runge-Kutta **method** that will be discussed in order to get the solutions in this study.

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