Analysis method of the stationary boundary-value problem of finite-amplitude wave according to Green formula.

We discuss the existence of solution for the fully fourth-order boundary value problemu(4)=f(t,u,u′,u′′,u′′′),0≤t≤1,u(0)=u(1)=u′′(0)=u′′(1)=0. A growth condition onfguaranteeing the existence of solution is presented. The discussion is based on the Fourier analysis method and Leray-Schauder fixed point theorem.

Download Full-text

A Boundary Value Problem with Multivariables Integral Type Condition for Parabolic Equations

Journal of Applied Mathematics and Stochastic Analysis ◽

10.1155/2009/975601 ◽

2009 ◽

Vol 2009 ◽

pp. 1-13

Author(s):

A. L. Marhoune ◽

F. Lakhal

Keyword(s):

Boundary Value Problem ◽

Parabolic Equations ◽

Continuous Dependence ◽

A Priori Estimates ◽

Boundary Value ◽

A Priori ◽

Analysis Method ◽

Type Condition ◽

Integral Type ◽

Priori Estimates

We study a boundary value problem with multivariables integral type condition for a class of parabolic equations. We prove the existence, uniqueness, and continuous dependence of the solution upon the data in the functional wieghted Sobolev spaces. Results are obtained by using a functional analysis method based on two-sided a priori estimates and on the density of the range of the linear operator generated by the considered problem.

Download Full-text

Three new approaches for solving a class of strongly nonlinear two-point boundary value problems

Boundary Value Problems ◽

10.1186/s13661-021-01536-3 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Monireh Nosrati Sahlan ◽

Hojjat Afshari

Keyword(s):

Integral Equation ◽

Boundary Value Problem ◽

Homotopy Analysis Method ◽

Boundary Value ◽

Scaling Functions ◽

Point Boundary ◽

Analysis Method ◽

Linearization Technique ◽

Strongly Nonlinear ◽

Homotopy Analysis

AbstractThree new and applicable approaches based on quasi-linearization technique, wavelet-homotopy analysis method, spectral methods, and converting two-point boundary value problem to Fredholm–Urysohn integral equation are proposed for solving a special case of strongly nonlinear two-point boundary value problems, namely Troesch problem. A quasi-linearization technique is utilized to reduce the nonlinear boundary value problem to a sequence of linear equations in the first method. Second method is devoted to applying generalized Coiflet scaling functions based on the homotopy analysis method for approximating the numerical solution of Troesch equation. In the third method we use an interesting technique to convert the boundary value problem to Urysohn–Fredholm integral equation of the second kind; afterwards generalized Coiflet scaling functions and Simpson quadrature are employed for solving the obtained integral equation. Introduced methods are new and computationally attractive, and applications are demonstrated through illustrative examples. Comparing the results of the presented methods with the results of some other existing methods for solving this kind of equations implies the high accuracy and efficiency of the suggested schemes.

Download Full-text

Homotopy Analysis Method for Boundary-Value Problem of Turbo Warrant Pricing under Stochastic Volatility

Abstract and Applied Analysis ◽

10.1155/2013/682524 ◽

2013 ◽

Vol 2013 ◽

pp. 1-5 ◽

Cited By ~ 2

Author(s):

Hoi Ying Wong ◽

Mei Choi Chiu

Keyword(s):

Boundary Value Problem ◽

Stochastic Volatility ◽

Homotopy Analysis Method ◽

Boundary Value ◽

Asset Price ◽

Analysis Method ◽

Over The Counter ◽

Homotopy Analysis ◽

Financial Derivative ◽

Pde Framework

Turbo warrants are liquidly traded financial derivative securities in over-the-counter and exchange markets in Asia and Europe. The structure of turbo warrants is similar to barrier options, but a lookback rebate will be paid if the barrier is crossed by the underlying asset price. Therefore, the turbo warrant price satisfies a partial differential equation (PDE) with a boundary condition that depends on another boundary-value problem (BVP) of PDE. Due to the highly complicated structure of turbo warrants, their valuation presents a challenging problem in the field of financial mathematics. This paper applies the homotopy analysis method to construct an analytic pricing formula for turbo warrants under stochastic volatility in a PDE framework.

Download Full-text