Top PDF The numerical solution of boundary value problems in partial differential equations

(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*

BOUNDARY VALUE PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS WITH PIECEWlSE CONSTANT DELAY.. JOSEPH WIENER Department of Mathematics The University of Texas Pan American Edinburg, Texas.[r]

x nJ+l matrix and r is an nJ+l-vector Employing Euler's method to approximate the differential equation lala, this formulation of the discrete problem becomes This example will be used t[r]

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 boundary value problems for higher order nonlinear ordinary differential equations, which have important applications in Fluid Dynamics, can be found in [1,2,6- 9,13,20]. Semper [2] and Roos and Stynes [8] considered fourth order equations and applied a standard finite element method. Garland [3] has shown that uniform stability of discrete boundary value 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 [3] and defect corrections are available in the literature. Loghmani and Ahmadinia [5] have developed a numerical technique for solving singularly perturbed boundary value problems based on optimal control strategy by using B-spline functions and least square method. Also finite element method is reported in [6,7]. In [9], an iterative method is described. In [10,11,16,17,20], the authors have applied boundary value technique to find the numerical solution for singularly perturbed second order boundary value problems. Niederdrenk and Yserentant [12] considered convection diffusion type problems and derived conditions for the uniform stabil- ity of discrete and continuous problems. Feckan [13] considered higher order problems and his approach is based on the nonlinear analysis involving fixed point theory, Leray-Schauder theory etc. In [15], authors have given a brief survey on computational techniques for the different classes of singularly perturbed problems. Bawa [19] and Aziz and Khan [21] have solved second order singularly perturbed boundary value problem using spline technique. Shanthi and Ramanujam [22-25] have developed numerical methods for singularly perturbed higher order boundary value problems.

In this paper we consider the numerical method of characteristics for the numerical solution of initial value problems (IVPs) for quasilinear hyperbolic Partial Differential Equations, as well as the difference scheme Central Time Central Space (CTCS), Crank-Nicolson scheme, ω scheme and the method of characteristics for the numerical solution of initial and boundary value 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 partial differential equations 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 [6].

The boundary-value problems for singularly perturbed delay-differential equations 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 [2], presented a fourth order finite difference method for solving singularly per- turbed differential difference equations. H. S. Prasad and Y. N. Reddy [3] considered Differential Quadrature Method for finding the numerical solution of boundary-value problems 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.

Those papers include solution of partial differential equations [1], two-point boundary value problems [2], integro-differential equations [3], second-kind integral equations [5], Fredho[r]

In studies of solutions of various types of nonlinear boundary value problems for ordinary differential equations 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 boundary value problems for ordinary and functional-differential equations. We refer, e.g., to the books [3-5], the articles [6-12], and the series of survey articles [13] 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-differential equations. 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 boundary value problems of Fredholm functional integro-differential equations.

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.

Three-point boundary value problems for ordinary differential equations by matching solutions, Nonlinear Anal.. Exlstence-unlqueness theorems for three-polnt.[r]

Numerical method is a substantial aspect in solving initial value problems in ordinary differential equations where the problems cannot be solved or difficult to obtain analytically. The numerical solutions of first order initial value problems have caught much attention recently; a new numerical scheme for the solution of initial value problems in ordinary differential equations was developed [10]. An integrator was also developed in [11] by representing the theoretical solution to initial value problems 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 boundary value problem for a system of partial differential equations 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-differential equations arise in connection with solving some special type of mixed boundary value problems 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. [5] used artificial neural networks (ANN) for solving ordinary differential equations and partial differential equations for both boundary value and initial value problems. Canh and Cong [6] 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 [7] used a modified neural network to solve the Berger’s equation in one-dimen- sional quasilinear partial differential equation.