(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*

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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]

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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]

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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**.

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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.

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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].

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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-

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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.

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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]

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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.

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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**.

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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.

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

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[.]

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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.

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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.

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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

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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.

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