Nonlinear parabolic problems in unbounded domains

In this paper, an approximate solution of non-linear parabolic partial differential equation is obtained for a non-uniform mesh. The scheme for partial differential equation subject to Neumann boundary is based on cubic B-spline collocation method. Modified cubic B-splines are proposed over non-uniform mesh to deal with the Dirichlet boundary conditions. This scheme produces a system of first order ordinary differential equations. This system is solved by Crank Nicholson method. The stability is also discussed using Von Neumann stability analysis. The accuracy and efficiency of the scheme is shown by numerical experiments. We have compared the approximate solutions with that in the literature.

The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential. The paper deals with the solutions of ordinary differential semi-linear equa- tion, the coefficients of which depend on several real parameters. If the coefficient is chosen so that the equation does not contain the first-order derivative of the unknown function, it will be the case of Emden - Fowler equation. Asymp- totic behavior of Emden - Fowler equation solutions at infinity is described in the book of Richard Bellman. The equations with the first-order derivative, considered in this work, erase in some problems for elliptic partial differential equations in unbounded domains. The sign of the coefficient in first-order deriva- tive term essentially influences on the description of solutions. Partly the result of this paper can be obtained from the works of I.T. Kiguradze. In present work we use lemmas about the behavior of solutions of the linear equations with a strongly (weakly) increasing potential.

Solutions of linear elliptic partial differential equations in unbounded domains can be represented by boundary potentials if they satisfy certain conditions at infinity. These radiation conditions depend on the fundamental solution chosen for the integral representation. We prove some basic results about radiation conditions in a rather general framework. Fundamental solutions G are considered that are defined only on the complement of a compact set. It turns out, however, and we present examples for this, that the more interesting results only hold if G is defined on all of ℝn or if it is a Green function for an exterior boundary value problem. © 1997 by B.G. Teubner Stuttgart-John Wiley & Sons, Ltd.

Risk Aversion and the Dynamics of Optimal Liquidation Strategies in Illiquid Markets

In this work, we explore a methodology to compute the empirical eigenfunctions for the order-reduction of nonlinear parabolic partial differential equations (PDEs) system with time-varying domain. The idea behind this method is to obtain the mapping functional, which relates the time-evolution scalar physical property solution ensemble of the nonlinear parabolic PDE with the time-varying domain to a fixed reference domain, while preserving space invariant properties of the raw solution ensemble. Subsequently, the Karhunen-Lo'eve decomposition is applied to the solution ensemble with fixed spatial domain resulting in a set of optimal eigenfunctions that capture the most energy of data. Further, the low dimensional set of empirical eigenfunctions is mapped (“pushed-back”) on the time-varying domain by an appropriate mapping resulting in the basis for the construction of the reduced-order model of the parabolic PDEs with time-varying domain. Finally, this methodology is applied in the representative cases of calculation of empirical eigenfunctions in the case of one and two dimensional model of nonlinear reaction-diffusion parabolic PDE systems with analytically defined domain evolutions. In particular, the design of both mappings which relate the raw data and function spaces transformations from the time-varying to time-invariant domain are designed to preserve dynamic features of the scalar physical property and we provide comparisons among reduced and high order fidelity models.

The aim of this paper is to extend the application of the Group Explicit method [1, 2] to the numerical solution of a non-linear parabolic partial differential equation of second order. The method was tested out on Burger's equation for various initial and boundary conditions. It can be seen that the method is accurate and comparable to existing finite difference methods.

Etant donne un polynome Φ de la forme Φ(α) = α + ∑2≤j≤m aj αk=1j respectant Φ'(1) > 0, nous demontrons que l’evolution, sur une echelle diffusive, de la densite empirique des processus d’exclusion sur ${\mathbb{T}}^{d}$, dont les conductances sont donnees par une classe speciale de fonctions W, est decrite par l’unique solution faible de l’equation aux derivees partielles parabolique : ∂tρ=∑d ∂xk ∂Wk Φ(ρ). Nous derivons egalement certaines proprietes de l’operateur ∑k=1d∂xk ∂Wk.

A finite‐difference method is developed for numerical solution of parabolic partial differential equations. This technique is explicit and stable. It is shown that the present method is more accurate and faster, in terms of computer time, than the Crank‐Nicholson method. A method of handling nonlinear problems is also presented. Two examples are given to illustrate the present technique. The first problem is a linear diffusion equation. The second problem deals with two simultaneous nonlinear parabolic partial differential equations with Neumann boundary conditions describing the steady state of a packed‐bed catalytic reactor with radial mixing.

An implicit compact scheme is proposed to approximate the solution of parabolic partial differential equations (PDEs) of Burgers’ type in two dimensions. These nonlinear PDEs are essential because they describe various mechanisms in engineering and physics. The nonlinear convective and advective processes are discretized with high-order accuracy on an arbitrary grid, which results in a family of high-resolution discrete replacements of given PDEs. The essence of the new scheme lies in its compact character and two-level single-cell discretization, so that one discrete equation leads to the accuracy of orders three or four, depending upon the choice of the grid network. The scheme is used for solving celebrated nonlinear PDEs, such as the nondegenerate convection–diffusion equation, the generalized Burgers–Huxley equation, the Buckley–Leverett equation, and the Burgers–Fisher equation. Many computational results are presented to demonstrate the high-resolution character of the newly proposed scheme.

We investigate the existence and Borel summability of formal power series in one variable with holomorphic coefficients, solutions of nonlinear partial differential equations with shrinkings having non-Kowalewski type in the complex domain. Sufficient conditions are given in terms of the shape of the functional partial differential equations and initial conditions. The existence and asymptotic behavior of local sectorial holomorphic solutions of these nonlinear functional partial differential equations are obtained as a by-product. We apply the results to the study of asymptotic the behavior of solutions of some advanced-argument partial differential equations in a neighborhood of infinity.

In this paper, the weighted average-based differential quadrature method is presented for solving one-dimensional homogeneous first-order non-linear parabolic partial differential equation. First, the given solution domain is discretized by using uniform discretization grid point. Next, by using Taylor series expansion we obtain central finite difference discretization of the partial differential equation involving with temporal variable associated with weighted average of partial derivative concerning spatial variable. From this, we obtain the system of nonlinear ordinary differential equations and it is linearized by using the quasilinearization method. Then by using the polynomial-based differential quadrature method for approximating derivative involving with spatial variable at specified grid point, we obtain the system of linear equation. Then they obtained linear system equation is solved by using the LU matrix decomposition method. To validate the applicability of the proposed method, two model examples are considered and solved at each specific grid point on its solution domain. The stability and convergent analysis of the present method is worked by supported the theoretical and mathematical statements and the accuracy of the solution is obtained. The accuracy of the present method has been shown in the sense of root mean square error norm and maximum absolute error norm and the local behavior of the solution is captured exactly. Numerical versus exact solutions and behavior of maximum absolute error between them have been presented in terms of graphs and the corresponding root mean square error norm and maximum absolute error norm presented in tables. The present method approximates the exact solution very well and it is quite efficient and practically well suited for solving the non-linear parabolic equation. The numerical result presented in tables and graphs indicates that the approximate solution is in good agreement with the exact solution.

Exponential second kind Chebyshev approximation for differential equations in unbounded domains, with applications

This paper is concerned with the asymptotic behavior of the solution, as $\varepsilon \to 0^ + $ or z tends to infinity in the complex z-plane, of the differential equation \[\varepsilon^n u^{(n)} (z) - \varepsilon ^{n - 1} p_{n - 1} (z) - \varepsilon ^{n - 2} p_{n - 2} (z)u^{(n - 2)} (z) - \cdots - p_0 (z)u(z) = 0,\] where $p_i (z)$, $i = 0,1, \cdots ,n - 1$, are polynomials in z with some restrictions on their degrees. A series of transformations of the dependent variable transforms the equation into a convenient almost diagonal system. Then, asymptotic series solutions are found by iterative solution of an equivalent integral equation. A careful study of the regions of validity in the z-plane of the asymptotic series shows that for every sufficiently narrow sector a full fundamental system of n asymptotically known solutions can be found.

In this paper we compare some families of weigthted Sobolev spaces which are commonly used for solving partial differential equations in unbounded domains. The first result isan identity between two particular spaces. The second result is another identity whichgeneralises partially the first one.

The problem of pricing European options based on multiple assets with transaction costs is considered. These options include, for example, quality options and options on the minimum of two or more risky assets. The value of these options is the solution of a nonlinear parabolic partial differential equation subject to a final condition given by the payoff function associated with the option. A computationally efficient method to solve this final-value problem is proposed. This method is based on an asymptotic expansion of the required solution with respect to the parameters related to the transaction costs followed by the numerical solution of the linear partial differential equations obtained at each order in perturbation theory. The numerical solution of these linear problems involves an implicit finite-difference scheme for the parabolic equation and the use of the fast Fourier sine transform to solve the resulting elliptic problems. Numerical results obtained on test problems with the method proposed here are shown and discussed.