scholarly journals Quadratic Spline Wavelets for Sparse Discretization of Jump–Diffusion Models

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 999
Author(s):  
Dana Černá
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Jump Diffusion ◽  
Diffusion Models ◽  
Dirichlet Boundary Conditions ◽  
Quadratic Spline ◽  
Vanishing Moments ◽  
Bounded Interval ◽  
Spline Wavelets ◽  
The Galerkin Method

This paper is concerned with a construction of new quadratic spline wavelets on a bounded interval satisfying homogeneous Dirichlet boundary conditions. The inner wavelets are translations and dilations of four generators. Two of them are symmetrical and two anti-symmetrical. The wavelets have three vanishing moments and the basis is well-conditioned. Furthermore, wavelets at levels i and j where i - j > 2 are orthogonal. Thus, matrices arising from discretization by the Galerkin method with this basis have O 1 nonzero entries in each column for various types of differential equations, which is not the case for most other wavelet bases. To illustrate applicability, the constructed bases are used for option pricing under jump–diffusion models, which are represented by partial integro-differential equations. Due to the orthogonality property and decay of entries of matrices corresponding to the integral term, the Crank–Nicolson method with Richardson extrapolation combined with the wavelet–Galerkin method also leads to matrices that can be approximated by matrices with O 1 nonzero entries in each column. Numerical experiments are provided for European options under the Merton model.

scholarly journals Convergence of approximate solution of mixed Hammerstein type integral equations

2018 ◽  
Vol 38 (2) ◽  
pp. 61-74
Author(s):  
Monireh Nosrati Sahlan
Keyword(s):  
Galerkin Method ◽  
Integral Equations ◽  
Basis Functions ◽  
Computational Method ◽  
Operational Matrices ◽  
B Spline ◽  
Spline Wavelets ◽  
Type Integral ◽  
The Galerkin Method ◽  
Dual Functions

In the present paper, a computational method for solving nonlinear Volterra-Fredholm Hammerestein integral equations is proposed by using compactly supported semiorthogonal cubic B-spline wavelets as basis functions. Dual functions and Operational matrices of B-spline wavelets via Galerkin method are utilized to reduce the computation of integral equations to some algebraic system, where in the Galerkin method dual of B-spline wavelets are applied as weighting functions. The method is computationally attractive, and applications are demonstrated through illustrative examples.

A Combined Fourier Series–Galerkin Method for the Analysis of Functionally Graded Beams

2004 ◽  
Vol 71 (3) ◽  
pp. 421-424 ◽  
Author(s):  
H. Zhu ◽  
B. V. Sankar
Keyword(s):  
Fourier Series ◽  
Differential Equations ◽  
Galerkin Method ◽  
Functionally Graded ◽  
Shear Stresses ◽  
Functionally Graded Beam ◽  
Fourier Series Method ◽  
Bending Stresses ◽  
The Galerkin Method ◽  
Graded Structures

The method of Fourier analysis is combined with the Galerkin method for solving the two-dimensional elasticity equations for a functionally graded beam subjected to transverse loads. The variation of the Young’s modulus through the thickness is given by a polynomial in the thickness coordinate and the Poisson’s ratio is assumed to be constant. The Fourier series method is used to reduce the partial differential equations to a pair of ordinary differential equations, which are solved using the Galerkin method. Results for bending stresses and transverse shear stresses in various beams show excellent agreement with available exact solutions. The method will be useful in analyzing functionally graded structures with arbitrary variation of properties.

scholarly journals Solving nonlinear boundary value problems by the Galerkin method with sinc functions

Open Physics ◽  
2015 ◽  
Vol 13 (1) ◽  
Author(s):  
Sertan Alkan ◽  
Aydin Secer
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Nonlinear Boundary ◽  
Nonlinear Differential Equations ◽  
Nonlinear Problems ◽  
Approximate Solutions ◽  
Nonlinear Boundary Value Problems ◽  
Sinc Approximation ◽  
Sinc Functions ◽  
The Galerkin Method

AbstractIn this paper, the sinc-Galerkin method is used for numerically solving a class of nonlinear differential equations with boundary conditions. The importance of this study is that sinc approximation of the nonlinear term is stated as a new theorem. The method introduced here is tested on some nonlinear problems and is shown to be a very efficient and powerful tool for obtaining approximate solutions of nonlinear ordinary differential equations.

Dynamics of a Cantilever Beam With Attached Pendulum

2001 ◽  
Author(s):  
Danuta Sado
Keyword(s):  
Differential Equations ◽  
Galerkin Method ◽  
Cantilever Beam ◽  
Weakly Nonlinear ◽  
Second Mode ◽  
Linear Viscoelastic ◽  
Nonlinear Pendulum ◽  
Analysis Of Dynamics ◽  
The Galerkin Method ◽  
Euler Bernoulli Beam

Abstract This work draws attention to the to the analysis of dynamics of a nonlinear coupled cantilever beam-pendulum oscillator. Dynamical systems of this type have important technical applications, because many mechanical components consist of linear or weakly nonlinear continuos substructures such as beam coupled to nonlinear oscillators. The present paper is a continuation of the author’s previous work where in applying the Galerkin method the modal series was truncated at the first mode. In this work it is assumed that the cantilever beam behaves like an Euler-Bernoulli beam and to its end pendulum is attached. The integro-differential equations are transformed into an ordinary differential equations with the use of Galerkin procedure with beam functions. In this study, in applying the Galerkin method the modal series was truncated at the second mode. Next these equations were solved numerically and there was studied the effect of the internal friction on energy transfer in a coupled structure that consist of a linear viscoelastic beam supporting at its tip a nonlinear pendulum.

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