(a) The problem In this olass Is oisaon's equation In two or more variables> (and the associated Laplace*s equation;* 'hen this equation ooours In physical problems* It usually desorlbes non-transient phenomena* i*e. phenomena independent of time* bus* no variable is time-like* and Xp 1 « 1>2**»*p* are space like variables* A properly posed proolem* In the sense that a solution (possibly non-unique) exists* can be shown to require boundary conditions on the boundary d of a dosed region 0* in £ space* The solution is then to be found in D* Llnoe the order of A Is two* It can be shown that only one boundary operator N* of degree one in the variables x^* 1 « 1* • ••»>» ia necessary* The equation which is considered is* therefore*
D = ( 0, 1 ) , D = [ 0, 1 ] and u eC 4 ( D ) \ C 2 ( D ) . ( 1.2 ) Analytical and numerical treatment of these equations have drawn much attention of many authors [5,22- 25]. The analytical treatment of singularly perturbed boundaryvalueproblems for higher order nonlinear ordinary differentialequations, which have important applications in Fluid Dynamics, can be found in [1,2,6- 9,13,20]. Semper  and Roos and Stynes  considered fourth order equations and applied a standard finite element method. Garland  has shown that uniform stability of discrete boundaryvalue problem follows from uniform stability of the discrete initial value problem and uniform consistency of the scheme. Some re- sults connected with the exponentially fitted higher order differences with identity expansion method  and defect corrections are available in the literature. Loghmani and Ahmadinia  have developed a numerical technique for solving singularly perturbed boundaryvalueproblems based on optimal control strategy by using B-spline functions and least square method. Also finite element method is reported in [6,7]. In , an iterative method is described. In [10,11,16,17,20], the authors have applied boundaryvalue technique to find the numericalsolution for singularly perturbed second order boundaryvalueproblems. Niederdrenk and Yserentant  considered convection diffusion type problems and derived conditions for the uniform stabil- ity of discrete and continuous problems. Feckan  considered higher order problems and his approach is based on the nonlinear analysis involving fixed point theory, Leray-Schauder theory etc. In , authors have given a brief survey on computational techniques for the different classes of singularly perturbed problems. Bawa  and Aziz and Khan  have solved second order singularly perturbed boundaryvalue problem using spline technique. Shanthi and Ramanujam [22-25] have developed numerical methods for singularly perturbed higher order boundaryvalueproblems.
In this paper we consider the numerical method of characteristics for the numericalsolution of initial valueproblems (IVPs) for quasilinear hyperbolic PartialDifferentialEquations, as well as the difference scheme Central Time Central Space (CTCS), Crank-Nicolson scheme, ω scheme and the method of characteristics for the numericalsolution of initial and boundaryvalue prob- lems for the one-dimension homogeneous wave equation. The initial deriva- tive condition is approximated by different second order difference quotients in order to examine which gives more accurate numerical results. The local truncation error, consistency and stability of the difference schemes CTCS, Crank-Nicolson and ω are also considered.
Meshless methods (MMs) are the next generation of computational methods development which are expected to be premier to the conventional grid-based FDM and FEM in many applications. The objectives of the MMs is to eliminate part of the difficulties associated with the accuracy and stable numerical solutions for partialdifferentialequations or integral equations with all kinds of boundary conditions without using any mesh to solve that problems. One of the main idea in these meshless methods is to modify the internal structure of mesh-based FEM and FDM to become more versatile and adaptive .
The boundary-valueproblems for singularly perturbed delay-differentialequations arise in various practical problems in biomechanics and physics such as in varia- tional problem in control theory. These problems mainly depend on a small positive parameter and a delay pa- rameter in such a way that the solution varies rapidly in some parts of the domain and varies slowly in some other parts of the domain. Moreover, this class of prob- lems possess boundary layers, i.e. regions of rapid change in the solution near one of the boundary points. There is a wide class of asymptotic expansion methods available for solving the above type problems. But there can be difficulties in applying these asymptotic expan- sion methods, such as finding the appropriate asymptotic expansions in the inner and outer regions, which are not routine exercises but require skill, insight and experi- mentation. The numerical treatment of singularly per- turbed problems present some major computational dif- ficulties and in recent years a large number of special- purpose methods have been proposed to provide accurate numerical solutions [1-5] by Kadalbajoo. This type of problem has been intensively studied analytically and it is known that its solution generally has a multiscale character; i.e. it features regions called “boundary lay- ers” where the solution varies rapidly. And these equa-
The problems in which the highest order derivative term is multiplied by a small parameter are known to be per- turbed problems and the parameter is known as the per- turbation parameter. A singularly perturbed differential- difference equation is an ordinary differential equation in which the highest derivative is multiplied by a small pa- rameter and involving at least one delay or advance term. Recently by constructing a special type of mesh, so that the term containing delay lies on nodal points after discretiza- tion R. N. Rao, P. P. Chakravarthy , presented a fourth order finite difference method for solving singularly per- turbed differential difference equations. H. S. Prasad and Y. N. Reddy  considered Differential Quadrature Method for finding the numericalsolution of boundary-valueproblems for a singularly perturbed differential-difference equation of mixed type. In recent papers [4-8] the terms negative or left shift and positive or right shift have been used for delay and advance respectively.
In studies of solutions of various types of nonlinear boundaryvalueproblems for ordinary differentialequations side by side with numerical methods, it is often used an appropriate technique based upon some types of successive approximations con- structed in analytic form. This class of methods includes, in particular, the approach suggested at first in [1,2] for investigation of periodic solutions. Later, appropriate ver- sions of this method were developed for handling more general types of nonlinear boundaryvalueproblems for ordinary and functional-differentialequations. We refer, e.g., to the books [3-5], the articles [6-12], and the series of survey articles  for the related references.
Mathematical modelling of real-life problems usually results in functional equations, of various types appear in many applications that arise in the fields of mathematical analysis, nonlinear functional analysis, mathematical physics, and engineering. An interesting feature of functional integral equations is their role in the study of many problems of functional integro-differentialequations. Several different techniques were proposed to study the existence of solutions of the functional integral equations in appropriate function spaces. Although all of these techniques have the same goal, they differ in the function spaces and the fixed point theorems to be applied. Consider the following boundaryvalueproblems of Fredholm functional integro-differentialequations.
Abstract This paper is devoted to the study of establishing suﬃcient conditions for existence and uniqueness of positive solution to a class of non-linear problems of fractional dif- ferential equations. The boundary conditions involved Riemann-Liouville fractional order derivative and integral. Further, the non-linear function f contain fractional order derivative which produce extra complexity. Thank to classical ﬁxed point the- orems of nonlinear alternative of Leray-Schauder and Banach Contraction principle, suﬃcient conditions are developed under which the proposed problem has at least one solution. An example has been provided to illustrate the main results.
Numerical method is a substantial aspect in solving initial valueproblems in ordinary differentialequations where the problems cannot be solved or difficult to obtain analytically. The numerical solutions of first order initial valueproblems have caught much attention recently; a new numerical scheme for the solution of initial valueproblems in ordinary differentialequations was developed . An integrator was also developed in  by representing the theoretical solution to initial valueproblems by an interpolating function which maybe linear or nonlinear.
BOUNDARY VALUE PROBLEHS FOR STOCHASTIC DIFFERENTIAL EQUATIONS Thesis by Thomas 1 Tilliam HacDm rell In Partial Fulfillment of the Requirements For the Degree of Doctor of Philosophy California Institu[.]
Abstract—We construct an explicit solution for a boundaryvalue problem for a system of partialdifferentialequations which describes small linearized motions of three-dimensional stratified flows in the half-space. For large values of t, we obtain uniform asymptotical decompositions of the solutions on an arbitrary compact in the half-space. In the vicinity of the boundary plane, we establish the asymptotical properties of the boundary layer type: we can observe a worsening of the decay in the approximation to the bottom. The results can be used in the meteorological modelling of water flows near the bottom of the Ocean, as well as Atmosphere flows near the Earth surface.
uniqueness results for the problem are provided and proved. Two monotone sequences of upper and lower solutions which converge uniformly to the unique solution of the problem are constructed using the method of lower and upper solutions. Sufficient numerical examples are discussed to corroborate the theory presented herein.
of problems concerning conduction and radiation. Also singular integro-differentialequations arise in connection with solving some special type of mixed boundaryvalueproblems involving the two dimensional Laplace’s equation in the quarter plane. Here we applied Legendre wavelets for solving such singular integro-differential equation. The numerical results show that the method has good accuaracy. We suppose that φ is L 2 [ − 1, 1] function and also Holder continuous.
Motivated by Webb and Infante [, ] and Webb and Lan , who established new ex- istence results of positive solutions of a Hammerstein integral equation in an uniﬁed way, under some growth condition imposed on the nonlinear term, we obtain explicit ranges of values of λ, μ, and ζ with which the problem (P λ,μ,ζ ) has a positive solution and has
Lagaris, et al.  used artificial neural networks (ANN) for solving ordinary differentialequations and partialdifferentialequations for both boundaryvalue and initial valueproblems. Canh and Cong  presented a new technique for numerical calculation of viscoelastic flow based on the combination of neural networks and Brownian dynamics simulation or stochastic simulation technique (SST). Hayati and Karami  used a modified neural network to solve the Berger’s equation in one-dimen- sional quasilinear partialdifferential equation.