Monotone Iteration Process Research Articles

On the Solution of the Black–Scholes Equation Using Feed-Forward Neural Networks

This paper deals with a comparative numerical analysis of the Black–Scholes equation for the value of a European call option. Artificial neural networks are used for the numerical solution to this problem. According to this method, we approximate the unknown function of the option value using a trial function, which depends on a neural network solution and satisfies the given boundary conditions of the Black–Scholes equation. We consider some optimization methods, not examined in the standard literature, such as particle swarm optimization and the gradient-type monotone iteration process, to obtain the unknown parameters of the neural network. Numerical results show that this proposed version of neural network method obtains all data from the terminal value and boundary conditions with sufficient accuracy.

Simultaneous reconstruction of the source term and the surface heat transfer coefficient

We study the problem of identifying unknown source terms in an inverse parabolic problem, when the overspecified (measured) data are given in form of Dirichlet boundary condition u(0,t) = h(t) and , where is an arbitrarily prescribed subregion. The main goal here is to show that the gradient of cost functional can be expressed via the solutions of the direct and corresponding adjoint problems. We prove Hölder continuity of the cost functional and derive the Lipschitz constant in the explicit form via the given data. On the basis of the obtained results, we propose a monotone iteration process. Copyright © 2014 John Wiley & Sons, Ltd.

Identification of an unknown source term in a vibrating cantilevered beam from final overdetermination

Inverse problems of determining the unknown source term F(x, t) in the cantilevered beam equation utt = (EI(x)uxx)xx + F(x, t) from the measured data μ(x) ≔ u(x, T) or ν(x) ≔ ut(x, T) at the final time t = T are considered. In view of weak solution approach, explicit formulae for the Fréchet gradients of the cost functionals J1(F) = ‖u(x, T; w) − μ(x)‖20 and J2(F) = ‖ut(x, T; w) − ν(x)‖20 are derived via the solutions of corresponding adjoint (backward beam) problems. The Lipschitz continuity of the gradients is proved. Based on these results the gradient-type monotone iteration process is constructed. Uniqueness and ill-conditionedness of the considered inverse problems are analyzed.

Simultaneous determination of the source terms in a linear hyperbolic problem from the final overdetermination: weak solution approach

The problem of determining the pair ω := {F(x,t); f(t)} of source terms in the hyperbolic equation u tt = (k(x)u x ) x + F(x, t) and in the Neumann boundary condition k(0)u x (0, t) = f (t) from the measured data μ (x) := u(x, T) and/or v(x) := u t (x, t) at the final time t = T is formulated. It is proved that both components of the Frechet gradient of the cost functionals J 1 (ω) = ∥u(x, t; w) ― μ(x)∥ 2 0 and J 2 (ω) = ∥u t (x, T; w) ― v(x) ∥ 2 0 can be found via the solutions of corresponding adjoint hyperbolic problems. Lipschitz continuity of the gradient is derived. Unicity of the solution and ill-conditionedness of the inverse problem are analysed. The obtained results permit one to construct a monotone iteration process, as well as to prove the existence of a quasi-solution.

A fourth-order compact finite difference method for higher-order Lidstone boundary value problems

A compact finite difference method is proposed for a general class of 2 n th-order Lidstone boundary value problems. The existence and uniqueness of the finite difference solution is investigated by the method of upper and lower solutions, without any monotone requirement on the nonlinear term. A monotone iteration process is provided for solving the resulting discrete system efficiently, and a simple and easily verified condition is obtained to guarantee a geometric convergence of the iterations. The convergence of the finite difference solution and the fourth-order accuracy of the proposed method are proved. Numerical results demonstrate the high efficiency and advantages of this new approach.

Time-Delayed finite difference reaction-diffusion systems with nonquasimonotone functions

This paper is concerned with finite difference solutions of a system of reaction-diffusion equations with coupled nonlinear boundary conditions and time delays. The reaction functions and the boundary functions are not necessarily quasimonotone, and the time delays may appear in the reaction functions as well as in the boundary functions. The investigation is devoted to the finite difference system for both the time-dependent problem and its corresponding steady-state problem. Some monotone iteration processes for the finite difference systems are given, and the asymptotic behavior of the time-dependent solution in relation to the steady-state solution is discussed. The asymptotic behavior result leads to some local and global attractors of the time-dependent problem, including the convergence of the time-dependent solution to a unique steady-state solution. An application and some numerical results to an enzyme-substrate reaction-diffusion problem are given. All the results are directly applicable to parabolic-ordinary systems and to reaction-diffusion systems without time delays.

Asymptotic behavior of the numerical solutions of time-delayed reaction diffusion equations with non-monotone reaction term

This paper is concerned with the asymptotic behavior of the finite difference solutions of a class of nonlinear reaction diffusion equations with time delay. By introducing a pair of coupled upper and lower solutions, an existence result of the solution is given and an attractor of the solution is obtained without monotonicity assumptions on the nonlinear reaction function. This attractor is a sector between two coupled quasi-solutions of the corresponding “steady-state" problem, which are obtained from a monotone iteration process. A sufficient condition, ensuring that two coupled quasi-solutions coincide, is given. Also given is the application to a nonlinear reaction diffusion problem with time delay for three different types of reaction functions, including some numerical results which validate the theoretical analysis.

Monotone Finite Difference Schemes for Nonlinear Systems with Mixed Quasi-Monotonicity

Monotone finite difference schemes are proposed for nonlinear systems with mixed quasi-monotonicity. Two monotone iteration processes for the corresponding discrete problems are presented, which converge monotonically to the quasi-solutions of the discrete problems. The limits are the exact solutions under some conditions. A monotone finite difference scheme on uniform mesh with the accuracy of fourth order is constructed. The numerical results coincide with theoretical analysis.

Asymptotic behavior of the numerical solutions for a system of nonlinear integrodifferential reaction–diffusion equations

This paper is concerned with the asymptotic behavior of the finite difference solutions of a coupled system of nonlinear integrodifferential reaction–diffusion equations. The existence of the finite difference solution and the monotone iteration process for solving the finite difference system are given. This includes an existence-uniqueness-comparison theorem. From the monotone iteration process, an attractor of the numerical time-dependent solution is obtained. This attractor is a sector between the pair of coupled quasisolutions of the corresponding numerical steady-state problem, which are obtained from a monotone iteration process. A sufficient condition, ensuring that the two coupled quasisolutions coincide, is given. Also given is the application to a reaction–diffusion problem with three different types of reaction functions, including some numerical results which validate the theory analysis.

Numerical Analysis of Coupled Systems of Nonlinear Parabolic Equations

This paper is concerned with numerical solutions of a general class of coupled nonlinear parabolic equations by the finite difference method. Three monotone iteration processes for the finite difference system are presented, and the sequences of iterations are shown to converge monotonically to a unique solution of the system, including an existence-uniqueness-comparison theorem. A theoretical comparison result for the various monotone sequences and an error analysis of the three monotone iterative schemes are given. Also given is the convergence of the finite difference solution to the continuous solution of the parabolic boundary-value problem. An application to a reaction-diffusion model in chemical engineering and combustion theory is given.

Monotone iterations for numerical solutions of reaction-diffusion-convection equations with time delay

In this article we use the monotone method for the computation of numerical solutions of a nonlinear reaction-diffusion-convection problem with time delay. Three monotone iteration processes for a suitably formulated finite-difference system of the problem are presented. It is shown that the sequence of iteration from each of these iterative schemes converges from either above or below to a unique solution of the finite-difference system without any monotone condition on the nonlinear reaction function. An analytical comparison result among the three processes of iterations is given. Also given is the application of the iterative schemes to some model problems in population dynamics, including numerical results of a model problem with known analytical solution. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 339–351, 1998

Dynamics of Nonlinear Parabolic Systems with Time Delays

The dynamics of a coupled system of semilinear parabolic equations with discrete time delays is investigated using the method of upper and lower solutions. It is shown that if the reaction function in the system possesses a mixed quasimonotone property and the corresponding elliptic system has a pair of coupled upper and lower solutions then there is a monotone iteration process which yields a pair of quasisolutions of the elliptic system and the sector between the quasisolutions is an attractor of the delayed parabolic system. Under some additional conditions this sector is a global attractor and the solution of the parabolic system converges to a true solution of the elliptic system. The same conclusions are obtained for a coupled system of parabolic-ordinary equations with time delays. Applications are given to three model problems arising from ecology and nuclear engineering. These model problems possess multiple steady-state solutions and sufficient conditions are given to ensure the stability and instability of these solutions.

Enclosure of solutions of weakly nonlinear elliptic boundary value problems and their computation

An algorithm is presented to compute approximations as well as continuous bounds for solutions of weakly nonlinear elliptic boundary value problems. The given problem is majorized in some sense and the obtained new problem is solved by a finite element method. The finite element solution is computed by a monotone iteration process and at last transformed to a continuous (lower) bound for a solution. Convergence is proved and mesh refinement effects are discussed. Two numerical examples are given.

On the solution of the Ambartsumian-Chandrasekhar equation by monotone iteration processes

On the solution of the Ambartsumian-Chandrasekhar equation by monotone iteration processes

On the acceleration of the convergence of monotone iteration processes for solving systems of linear algebraic equations

On the acceleration of the convergence of monotone iteration processes for solving systems of linear algebraic equations