domain. Since the integration is in , they impose the requirement that the diﬀusion takes place only in . The individuals may not enter nor leave . This is the analogous of what is called **homogeneous** **Neumann** **boundary** **conditions** in the literature. One analyzed the existence and uniqueness of solutions for (1.4) and found an exponential convergence to the mean value of the initial condition.

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with **homogeneous** **boundary** **conditions** u(x, ) = u(x, ) = u(, t) = u(, t) = . The ana- lytical solution of this problem is u(x, t) = xt(x – )(t – ). The graph of exact solution is shown in Figure . The graphs of numerical solutions for m = , , in some nodes (x, t) ∈ [, ] × [, ] are shown in Figures , , . From Examples and Examples , it can be concluded that the numerical solutions approximate the exact solutions well as m grows.

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In this context, by using a diﬀerent approach with respect to [4], we are able to prove existence results for solutions of (1) subjected to very general **boundary** value **conditions** including, as particular cases, Dirichlet, periodic, Sturm-Liouville and **Neumann** prob- lems. Our results extend those in [4] both for the presence of the function a inside the diﬀerential operators and for the great generality of the structure on the **boundary** con- ditions. Finally, we also provide some examples of application of our results, in which the operator is not **homogeneous** and grows exponentially at inﬁnity.

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The paper is organized as follows. In the next section we study the properties of some integro-diﬀerential operators, which we then use throughout the paper. In Section we investigate the Dirichlet problem for a polyharmonic equation, making use of the ex- plicit form of the Green function found in [–]. Then, in the following section, reducing **Neumann** problem (), () to the considered Dirichlet problem, we give the necessary and suﬃcient solvability condition for problem (), () with **homogeneous** **boundary** condi- tions. In the same way we consider in Section the **Neumann** **boundary** value problem for the **homogeneous** equation with non-**homogeneous** **boundary** **conditions**. Finally, in Sec- tion we study problem (), () in the general case. To present the necessary and suﬃcient **conditions** for solvability, here we apply the Almansi formula for constructing solutions to the Dirichlet problem.

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Our method of choice for finding a proper solution decomposition into an interior part (arising from solutions of third-order problems) and layer parts is the method of asymptotic expansions. This approach can, for instance, be found in [4, 10, 11, 13], where it is applied to second-order problems. Roughly rephrasing the rationale of asymptotic expansions, we consider a reduced problem, where formally ε is set to 0 and certain **boundary** **conditions** are neglected, to construct solution parts that are independent of ε. The misfit of certain **boundary** **conditions** is then corrected by **boundary** layer terms. As an ansatz for the **boundary** layer terms, we choose functions that exponentially decay away from the **boundary**.

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The global existence and blow-up for nonlinear parabolic equations have been extensively investigated by many authors in the last decades (see [1–6] and the references therein). In recent years, many authors have also studied bounds for the blow-up time in nonlinear parabolic problems by using differential inequality techniques (see [7–12]). In particular, Payne et al. [13] considered the following semilinear heat equation with nonlinear **boundary** condition

Abstract. This work is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic program- ming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton- Jacobi-Bellman (HJB) equation with an oblique derivative **boundary** condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in fi- nance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled sto- chastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach.

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Levine, H., Zhang, Q.: The critical Fujita number for a semilinear heat equation in exterior domains with homogeneous Neumann boundary values.. Li, H., Wang, X., Nie, Y., He, H.: Asympto[r]

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Evidence from studies prior to Tuch et al. suggested a directional relationship between aesthetics and usability in which increased aesthetics equaled increased usability. However, Tuch et al. showed that under certain circumstances the direction of the relationship was reversed—lower usability equaled lower perceived aesthetics. This finding was new, and it demonstrated the importance of exploring the contingencies and **boundary** **conditions** of the specific effects of manipulations of aesthetics and usability. Tuch et al. pointed out that additional research was necessary to understand the directions of these effects. They also pointed out that their results differed from prior studies in that users’ perceptions of the aesthetics of an interface changed after experience with the interface. However, their exploration of that particular contingency was limited to two observations, one made immediately before and the other immediately after users’ one-time interaction with the system. Recognizing this limitation, Tuch et al. encouraged future research that further manipulated aesthetics and usability to observe which effects occur under which **conditions**. Research Questions and Hypotheses

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Abstract In the present paper, a numerical method is considered for solving one-dimensional heat equation subject to both **Neumann** and Dirichlet initial **boundary** **conditions**. This method is a combination of collocation method and radial basis functions (RBFs). The operational matrix of derivative for Laguerre-Gaussians (LG) radial basis functions is used to reduce the problem to a set of algebraic equations. The re- sults of numerical experiments are presented to conﬁrm the validity and applicability of the presented scheme.

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The classical panel method has been extensively used in external aero- dynamics to calculate ideal ow elds around moving vehicles or stationary structures in unbounded domains. However, the panel method, as a somewhat simpler implementation of the **boundary** element method, has rarely been employed to solve problems in closed complex domains. This paper aims at lling this gap and discusses the numerical solution of the Laplace equation in bounded domains via the numerical panel method. It is shown that the panel method is an ecient and accurate computational algorithm for the solution of this class of problems. Several test cases in heat conduction and internal ideal ow are presented to show that the numerical panel method can be used in closed domains regardless of the complexities in the geometry andor **boundary** **conditions**.

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This paper investigates the existence of solutions for prescribed variable exponent mean curvature impulsive system, with periodic-like, Dirichlet, and **Neumann** **boundary** value **conditions**, respectively. The proof of our main result is based upon the Leray-Schauder degree. The suﬃcient **conditions** for the existence of solutions are given.

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The above equations and **boundary** **conditions** were coded using the computer algebra system, Mathematica for both the quadratic and cubic temperature profiles. Two Reynolds numbers were considered, i.e., 1,000 and 500,000 as were three values of ′ , 1/2, 1/4 and 1/24. The results of the developed model were compared with those obtained using a computational fluid dynamics code.

over, the asymptotic behavior of these first curves was shown to depend on the supports of the weights. The case of the **Neumann** **boundary** **conditions** was considered later in [2] where, contrary to what happened in the Dirichlet case, the asymptotic behavior of the first curve did not depend on the supports of the weights.

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Abstract:In this paper, one-dimensional heat equation subject to both **Neumann** and Dirichlet initial **boundary** **conditions** is presented and a Homotopy Perturbation Method is utilized for solving the problem. Homotopy Perturbation Method provides continuous solution in contrast to finite difference method, which only provides discrete approximations. It is found that this method is a powerful mathematical tool and can be applied to a large class of linear and non linear problem in different fields of science, engineering and technology.

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Chapter 4 is devoted to the study of nonlinear scalar two-point **boundary** value problems where the corresponding linear **homogeneous** **boundary** value problem has a one dimensional solution space. Through use of the Lyapunov–Schmidt Procedure **conditions** are established to guarantee the existence of solutions to the these bound- ary value problems. We pay particular attention to second order equations subject to periodic **boundary** **conditions**. Results are obtained which significantly extend previ- ous work by Etheridge and Rodr´ıguez concerning the periodic behavior of nonlinear discrete dynamical systems [10].

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Equation (.) describes the diﬀusion of concentration of some Newtonian ﬂuids through porous media or the density of some biological species in many physical phe- nomena and combustion theories (see [, ]). The nonlinear **Neumann** **boundary** value condition (.) can be physically interpreted as the nonlinear radial law (see, e.g., [, ]).

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They have proved the existence of nontrivial weak so- lutions for this problem for every given h ∈ L p (Ω) un- der some assumptions on the function g. In the case of **Neumann** elliptic problem J.-P. Gossez and P. Omari, have considered in [2] the following problem

Another numerical technique for finding approxi- mate solutions of partial differential equation is the finite element method (FEM). The solution approach is based either on eliminating the differential equation completely or rendering the partial differential equation into an approximating system of ordering differential equation, which are then numerically integrated using standard techniques such as Euler's method, Runge- Kutta, etc. In solving partial differential equations, the primary challenge is to create an equation that approximates the equation under study, and is also numerically stable. This means that errors in the input and intermediate calculations do not accumulate and, therefore, do not cause the resulting output to be meaningless. In so doing, there are several ways, all of which have both advantages and disadvantages. The Finite Element Method is a good choice for solving partial differential equations over complicated domains (like cars and oil pipelines), when the domain changes (as during a solid state reaction with a moving **boundary**), when the desired precision varies over the entire domain, or when the solution lacks smoothness. For instance, in a frontal crash simulation it is possible to increase prediction accuracy in important areas like the front of the car and reduce it in its rear. The differences between FEM and FDM are:

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In this subsection, the performance of the EBNUM is tested on five problems, which include the Poisson equa- tion, Laplace equation subject to **Neumann** **boundary** **conditions**, Laplace equation subject to Dirichlet **boundary** **conditions**, Helmoltz equation, and the two-dimensional convection diffusion equation. In all the figures, the EBNUM is represented by uapprox and the exact solution is represented by uexact.

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