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The main contribution of this work is illustrated as follows: 1- The derivation for embedded singly diagonally implicit Runge-Kutta (SDIRK) method of fourth-order six stages in fifth-order seven stages is introduced to solve ordinary and delay differential equations. The stability region is presented and the numerical results are compared with the other existing methods. 2- Singly diagonally implicit Runge-Kutta Nystróm (SDIRKN) of third-order three stages embedded in fourth-order four stages is constructed. The stability region of the new method is presented and numerical results are compared with the same method of lower order. 3- A new singly diagonally implicit Runge-Kutta-Nystróm general (SDIRKNG) method of third-order embedded in fourth-order is derived to solve second order ordinary differential equations. Analysis the stability region of the new method is discussed and numerical results are presented.

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The numerical approximation of solutions of ordinarydifferential equations played an important role inNumerical Analysis and still continues to be anactive field of research. In this book we are mainlyconcerned with the numerical solution of thefirst-order system of nonlinear two-point boundaryvalue problems. We will focus on the problem ofsolving singular perturbation problems since this hasfor many years been a source of difficulty to appliedmathematicians, engineers and numerical analystsalike. Firstly, we consider deferred correctionschemes based on Mono-Implicit Runge-Kutta (MIRK) andLobatto formulae. As is to be expected, the schemebased on Lobatto formulae, which are implicit, ismore stable than the scheme based on MIRK formulaewhich are explicit. Secondly, we construct high orderinterpolants to provide the continuous extension ofthe discrete solution. It will consider theconstruction of both explicit and implicitinterpolants. The estimation of conditioning numbersis also discussed and used to develop mesh selectionalgorithms which will be appropriate for solvingstiff linear and nonlinear boundary value problems.

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The gas-kinetic methods for viscous flow simulations have attracted much attention and become mature during the past decade. Ever since they were originally invented for compressible gas flows, the gas-kinetic methods have been extended to various viscous flow problems with a large number of applications. This book not only provides an introduction to the gas-kinetic methods, but also presents some recent progress on these useful and powerful methods. Besides the original gas-kinetic finite volume scheme for the Euler and Navier-Stokes equations, the book also covers in detail the following extensions of the method: the high order gas-kinetic Runge-Kutta discontinuous Galerkin finite element method for Navier-Stokes equations, the multiscale gas-kinetic method for flows in near continuum regime including rarefied gas and microscale gas flows, the multiple temperature kinetic model for microscale flow problems. Plenty of numerical examples are exhibited to validate these methods. The book addresses researchers as well as graduate students and engineers interested in learning, using, or further developing the gas-kinetic methods.

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Majority of the current numerical simulation methods for time-dependent flows are limited to second order accuracy in time discretization. Moreover, usual simulation methods rely on explicit time discretization methods for which numerical stability of solution is generally guaranteed only with very small time step sizes. Major commercial CFD software programs provide options for implicit time advancing, but the accuracy is limited to second order in time. In this book a stable simulation method is proposed that can be used to achieve arbitrarily high order of accuracy in time advancement in simulation of time-dependent flows and heat transfer. The strategy is to combine the state-of-the-art mathematical tools with proven flow simulation algorithms to develop simulation techniques with higher-order accuracy. A special class of implicit Runge-Kutta methods is used in conjunction with SIMPLE algorithm. The proposed method is called SIMPLE DIRK method. This book will be helpful to university students and researchers who are involved in research and code development for simulation of time-dependent flows and heat transfer.

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A numerical analysis was performed as to investigate the heat transfer characteristics of an evaporating thin-film meniscus. A mathematical model was used in the formulation of a third order ordinary differential equation. This equation governs the evaporating thinfilm through use of continuity, momentum, energy equations and the Kelvin-Clapeyron model. This governing equation was treated as an initial value problem and was solved numerically using a Runge-Kutta technique. The numerical model uses varying thermophysical properties and boundary conditions such as channel width, applied superheat, accommodation coefficient and working fluid which can be tailored by the user. This work focused mainly on the effects of altering accommodation coefficient and applied superheat. A unified solution is also presented which models the meniscus to half channel width. The model was validated through comparison to literature values.

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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. The finite element method (FEM) (its practical application often known as finite element analysis (FEA)) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The solution approach is based either on eliminating the differential equation completely (steady state problems), or rendering the PDE into an approximating system of ordinary differential equations, which are then numerically integrated using standard techniques such as Euler''s method, Runge-Kutta, etc.

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The book deals with the basic concepts of fluid dynamics and heat, mass transfer behavior of Newtonian and non-Newtonian fluid past an extending surfaces. The considered subject is perceived by the researchers and post-doctoral researchers. In this book we introduces computational numerical tool for solving highly non-linear differential equations models that arise in flow of fluid, heat and mass transfer of both Newtonian/non-Newtonian fluid problem. Self similar solutions are obtained using the fourth-fifth order Runge-Kutta-Fehlberg method. The considered work has great importance in the field of science and technology. The considered problem is quite useful in guided missiles, rain erosion, uidization, atmospheric fallout, paint and aerosol spraying,lunar ash ows,and cooling of nuclear reactor.

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The purpose of this work is to determinate the approximate solutions of boundary value problems with conditions inside the interval (0,1) using collocation method with global B-splines functions of degree k(order k+1), orthogonal polynomials Chebyshev and combined methods with B-splines functions or C.C method and Runge-Kutta methods. We do a comparative study of these numerical methods, making their implementation of algorithms written in MATLAB 2011b, Maple 2014 and we was also concerned with the approximations errors and determine their implementations costs (run time and internal memory used).

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This book deals with the elementary aspects of flow, heat and mass transfer of boundary layer flow problems related to magneto hydrodynamic non- Newtonian nanofluid flows , with special emphasis on slip flow and melting heat transfer so that the subject is perceived by the post graduate students, researchers and post-doctoral researchers. This text also introduces numerical computational tools for solving differential equation models that arise in fluid flow, heat and mass- transfer of non-Newtonian fluids. This study is essentially numerical in character. By applying the similarity transformations, the system of non-linear partial differential equations are reduced into a set of non-linear ordinary differential equations. Obtained equations are then solved numerically using Runge-Kutta-Fehlberg-45 order method along with Shooting technique.

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