30,813 research outputs found

    Transformation Method for Solving Hamilton-Jacobi-Bellman Equation for Constrained Dynamic Stochastic Optimal Allocation Problem

    Full text link
    In this paper we propose and analyze a method based on the Riccati transformation for solving the evolutionary Hamilton-Jacobi-Bellman equation arising from the stochastic dynamic optimal allocation problem. We show how the fully nonlinear Hamilton-Jacobi-Bellman equation can be transformed into a quasi-linear parabolic equation whose diffusion function is obtained as the value function of certain parametric convex optimization problem. Although the diffusion function need not be sufficiently smooth, we are able to prove existence, uniqueness and derive useful bounds of classical H\"older smooth solutions. We furthermore construct a fully implicit iterative numerical scheme based on finite volume approximation of the governing equation. A numerical solution is compared to a semi-explicit traveling wave solution by means of the convergence ratio of the method. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index as an example of application of the method

    Time- or Space-Dependent Coefficient Recovery in Parabolic Partial Differential Equation for Sensor Array in the Biological Computing

    Get PDF
    This study presents numerical schemes for solving a parabolic partial differential equation with a time- or space-dependent coefficient subject to an extra measurement. Through the extra measurement, the inverse problem is transformed into an equivalent nonlinear equation which is much simpler to handle. By the variational iteration method, we obtain the exact solution and the unknown coefficients. The results of numerical experiments and stable experiments imply that the variational iteration method is very suitable to solve these inverse problems

    Numerical solution of a free boundary problem by interchanging dependent and independent variables

    Get PDF
    The classical problem of seepage of fluid through a porous dam is solved to illustrate a new approach to more general free boundary problems. The numerical method is based on the interchange of the dependent variable, representing velocity potential, with one of the independent space variables, which becomes the new variable to be computed. The need to determine the position of the whole of the free boundary in the physical plane is reduced to locating the position of the separation point on a fixed straight—line boundary in the transformed plane. An iterative algorithm approximates within each single loop both a finite-difference solution of the partial differential equation and the position of the free boundary. The separation point is located by fitting a 'parabolic tail' to the finite-difference solution

    Sensitivity analysis of the early exercise boundary for American style of Asian options

    Full text link
    In this paper we analyze American style of floating strike Asian call options belonging to the class of financial derivatives whose payoff diagram depends not only on the underlying asset price but also on the path average of underlying asset prices over some predetermined time interval. The mathematical model for the option price leads to a free boundary problem for a parabolic partial differential equation. Applying fixed domain transformation and transformation of variables we develop an efficient numerical algorithm based on a solution to a non-local parabolic partial differential equation for the transformed variable representing the synthesized portfolio. For various types of averaging methods we investigate the dependence of the early exercise boundary on model parameters

    Solution of hyperbolic bioheat transfer problems by numerical green's functions: the ExGA-linear θ method

    Get PDF
    This paper presents a time-domain formulation called Explicit Green's approach (ExGA) linear θ method for the solution of the bioheat equation. Starting from the hyperbolic bioheat equation, which includes the parabolic one as a special case, the linear method is incorporated into the standard ExGA time marching scheme. The numerical Green's function is firstly computed in the Laplace transform domain and then back-transformed to the time domain through the Stehfest inversion algorithm. The proposed formulation has the properties of stabilizing the results and suppressing numerical oscillations that appear in the presence of discontinuous solutions as assessed through the analysis of some bioheat transfer problems.

    Tsunami runup in U and V shaped bays

    Get PDF
    Thesis (M.S.) University of Alaska Fairbanks, 2014.Tsunami runup can be effectively modeled using the shallow water wave equations. In 1958 Carrier and Greenspan in their work "Water waves of finite amplitude on a sloping beach" used this system to model tsunami runup on a uniformly sloping plane beach. They linearized this problem using a hodograph type transformation and obtained the Klein-Gordon equation which could be explicitly solved by using the Fourier-Bessel transform. In 2011, Efim Pelinovsky and Ira Didenkulova in their work "Runup of Tsunami Waves in U-Shaped Bays" used a similar hodograph type transformation and linearized the tsunami problem for a sloping bay with parabolic cross-section. They solved the linear system by using the D'Alembert formula. This method was generalized to sloping bays with cross-sections parameterized by power functions. However, an explicit solution was obtained only for the case of a bay with a quadratic cross-section. In this paper we will show that the Klein-Gordon equation can be solved by a spectral method for any inclined bathymetry with power function for any positive power. The result can be used to estimate tsunami runup in such bays with minimal numerical computations. This fact is very important because in many cases our numerical model can be substituted for fullscale numerical models which are computationally expensive, and time consuming, and not feasible to investigate tsunami behavior in the Alaskan coastal zone, due to the low population density in this areaIntroduction -- Chapter 1. Description of the problem -- Chapter 2. Linearization of the system of shallow water equations -- 2.1. Method of characteristics -- 2.2. Change of variables -- 2.3. Boundary conditions, initial conditions and domain of Φ, the linearized shallow water equation -- 2.4. The limits of applicability of the hodograph transformation -- Chapter 3. Solution of the linearized shallow water equation (the equation (2.19)) -- 3.1. Laplace transformation -- 3.2. Solving the transformed equation -- 3.3. Inverse Laplace transformation -- 3.4. Obtaining the formula for the solution of the linearized shallow water equation -- 3.5. The formula for Φ with a different order of integration -- Chapter 4. Verification of the solution of the linearized equation, obtained by the Laplace transform, for particular cases -- 4.1. Case 1. Plane beach case -- 4.1.1. Method 1. Solving by Laplace transform -- 4.1.2. Method 2. Solving by Fourier-Bessel transform after Carrier-Greenspan -- 4.2. Case 2. An inclined bay with the parabolic cross-section -- 4.2.1. Method 1. Solving by Laplace transform -- 4.2.2. Method 2. Solving by D'Alembert method after Didenkulova and Pelinovsky -- Chapter 5. Relation of the shallow water problem to the wave equation in Rn space -- 5.1. Solution of the wave equation and its spherical mean -- 5.2. Shallow water problems related to the waves in odd-dimensional spaces -- Chapter 6. Sloping bay with the cross-section parameterized by z = cIy ²/₃, c > 0 -- 6.1. Derivation of Φ (λ, σ), the solution of the linearized shallow water equation (6.1) with given initial and boundary conditions -- 6.2. Comparison of the obtained solution Φ for the bay described by z = cIy ²/₃ with the solution obtained through the Laplace transform -- 6.3. Physical characteristics of a wave in the bay and partial derivatives of Φ (λ,σ) -- Chapter 7. Practical experiments -- Conclusion -- References -- Appendix -- Chapter A. Modified Bessel functions and their asymptotic formulas -- A.1. Modified Bessel functions -- A.2. The asymptotic behavior of the modified Bessel functions

    A Method of Verified Computations for Solutions to Semilinear Parabolic Equations Using Semigroup Theory

    Get PDF
    This paper presents a numerical method for verifying the existence and local uniqueness of a solution for an initial-boundary value problem of semilinear parabolic equations. The main theorem of this paper provides a sufficient condition for a unique solution to be enclosed within a neighborhood of a numerical solution. In the formulation used in this paper, the initial-boundary value problem is transformed into a fixed-point form using an analytic semigroup. The sufficient condition is derived from Banach\u27s fixed-point theorem. This paper also introduces a recursive scheme to extend a time interval in which the validity of the solution can be verified. As an application of this method, the existence of a global-in-time solution is demonstrated for a certain semilinear parabolic equation
    corecore