scholarly journals A semilinear Black and Scholes partial differential equation for valuing American options: approximate solutions and convergence

2004 ◽  
pp. 379-404 ◽  
Author(s):  
F. Benth ◽  
Kenneth Karlsen ◽  
K. Reikvam
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
American Options ◽  
Approximate Solutions ◽  
Partial Differential

ESTIMATES FOR APPROXIMATE SOLUTIONS TO A FUNCTIONAL DIFFERENTIAL EQUATION MODEL OF CELL DIVISION

2021 ◽  
pp. 1-20
Author(s):  
STEPHEN TAYLOR ◽  
XUESHAN YANG
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cell Division ◽  
Nonnegative Function ◽  
Approximate Solutions ◽  
Equation Model ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order ◽  
Functional Differential

Abstract The functional partial differential equation (FPDE) for cell division, $$ \begin{align*} &\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t))\\ &\quad = -(b(x,t)+\mu(x,t))n(x,t)+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t), \end{align*} $$ is not amenable to analytical solution techniques, despite being closely related to the first-order partial differential equation (PDE) $$ \begin{align*} \frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t), \end{align*} $$ which, with known $F(x,t)$ , can be solved by the method of characteristics. The difficulty is due to the advanced functional terms $n(\alpha x,t)$ and $n(\beta x,t)$ , where $\beta \ge 2 \ge \alpha \ge 1$ , which arise because cells of size x are created when cells of size $\alpha x$ and $\beta x$ divide. The nonnegative function, $n(x,t)$ , denotes the density of cells at time t with respect to cell size x. The functions $g(x,t)$ , $b(x,t)$ and $\mu (x,t)$ are, respectively, the growth rate, splitting rate and death rate of cells of size x. The total number of cells, $\int _{0}^{\infty }n(x,t)\,dx$ , coincides with the $L^1$ norm of n. The goal of this paper is to find estimates in $L^1$ (and, with some restrictions, $L^p$ for $p>1$ ) for a sequence of approximate solutions to the FPDE that are generated by solving the first-order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper.

scholarly journals Sinc-Galerkin method for solving hyperbolic partial differential equations

2018 ◽  
Vol 8 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Aydin Secer
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Exact Solutions ◽  
Galerkin Method ◽  
Hyperbolic Equations ◽  
Approximate Solutions ◽  
Equation System ◽  
Hyperbolic Partial Differential Equations ◽  
Partial Differential ◽  
Numerical Examples

In this work, we consider the hyperbolic equations to determine the approximate solutions via Sinc-Galerkin Method (SGM). Without any numerical integration, the partial differential equation transformed to an algebraic equation system. For the numerical calculations, Maple is used. Several numerical examples are investigated and the results determined from the method are compared with the exact solutions. The results are illustrated both in the table and graphically.

On the Approximate Solutions of Heat-Conduction Problems

1963 ◽  
Vol 85 (3) ◽  
pp. 203-207 ◽  
Author(s):  
Fazil Erdogan
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Approximate Solutions ◽  
Integral Transforms ◽  
Partial Differential ◽  
The Given

Integral transforms are used in the application of the weighted residual methods to the solution of problems in heat conduction. The procedure followed consists in reducing the given partial differential equation to an ordinary differential equation by successive applications of appropriate integral transforms, and finding its solution by using the weighted-residual methods. The undetermined coefficients contained in this solution are functions of transform variables. By inverting these functions the coefficients are obtained as functions of the actual variables.

scholarly journals On Semianalytical Study of Fractional-Order Kawahara Partial Differential Equation with the Homotopy Perturbation Method

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Muhammad Sinan ◽  
Kamal Shah ◽  
Zareen A. Khan ◽  
Qasem Al-Mdallal ◽  
Fathalla Rihan
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Solitary Wave ◽  
Perturbation Method ◽  
Fractional Order ◽  
Homotopy Perturbation Method ◽  
Nonlinear Partial Differential Equation ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Homotopy Perturbation

In this study, we investigate the semianalytic solution of the fifth-order Kawahara partial differential equation (KPDE) with the approach of fractional-order derivative. We use Caputo-type derivative to investigate the said problem by using the homotopy perturbation method (HPM) for the required solution. We obtain the solution in the form of infinite series. We next triggered different parametric effects (such as x, t, and so on) on the structure of the solitary wave propagation, demonstrating that the breadth and amplitude of the solitary wave potential may alter when these parameters are changed. We have demonstrated that He’s approach is highly effective and powerful for the solution of such a higher-order nonlinear partial differential equation through our calculations and simulations. We may apply our method to an additional complicated problem, particularly on the applied side, such as astrophysics, plasma physics, and quantum mechanics, to perform complex theoretical computation. Graphical presentation of few terms approximate solutions are given at different fractional orders.

Estimates for approximate solutions to a functional differential equation model of cell division

2021 ◽  
Vol 62 ◽  
pp. 469-488
Author(s):  
Stephen William Taylor ◽  
Xueshan Yang
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Cell Division ◽  
Nonnegative Function ◽  
Approximate Solutions ◽  
Equation Model ◽  
Order Partial Differential Equation ◽  
Partial Differential ◽  
First Order ◽  
Functional Differential

The functional partial differential equation (FPDE) for cell division, \[\frac{\partial}{\partial t}n(x,t)+\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)\] \[+b(\alpha x,t)\alpha n(\alpha x,t)+b(\beta x,t)\beta n(\beta x,t),\] is not amenable to analytical solution techniques, despite being closely related to the first order partial differential equation (PDE) \[\frac{\partial}{\partial t}n(x,t) +\frac{\partial}{\partial x}(g(x,t)n(x,t)) = -(b(x,t)+\mu(x,t))n(x,t)+F(x,t),\] which, with known \(F(x,t)\), can be solved by the method of characteristics. The difficulty is due to the advanced functional terms \(n(\alpha x,t)\) and \(n(\beta x,t)\), where \(\beta \ge 2 \ge\alpha \ge 1\), which arise because cells of size \(x\) are created when cells of size  \(\alpha x\) and \(\beta x\) divide. The nonnegative function, \(n(x,t)\), denotes the density of cells at time \(t\) with respect to cell size \(x\). The functions \(g(x,t)\), \(b(x,t)\) and \(\mu(x,t)\) are, respectively, the growth rate, splitting rate and death rate of cells of size \(x\). The total number of cells, \(\int_{0}^{\infty}n(x,t)dx\), coincides with the \(L^1\) norm of \(n\). The goal of this paper is to find estimates in \(L^1\) (and, with some restrictions, \(L^p\) for \(p>1\)) for a sequence of approximate solutions to the FPDE that are generated by solving the first order PDE. Our goal is to provide a framework for the analysis and computation of such FPDEs, and we give examples of such computations at the end of the paper. doi:10.1017/S1446181121000055

scholarly journals An Improved Approach for Solving Partial Differential Equation Based on Reproducing Kernel Method

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Mingjing Du
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Present Method ◽  
Reproducing Kernel ◽  
Numerical Solutions ◽  
Kernel Method ◽  
Approximate Solutions ◽  
Partial Differential ◽  
Reproducing Kernel Method ◽  
First Time

The traditional reproducing kernel method (TRKM) cannot obtain satisfactory numerical results for solving the partial differential equation (PDE). In this study, for the first time, the abovementioned problems are solved by adaptive piecewise interpolation reproducing kernel method (APIRKM) to obtain the exact and approximate solutions of partial differential equations by means of series expansion using reconstructed kernel function. The highlight of this paper is to obtain more accurate approximate solution and save more time through adaptive discovery. Numerical solutions of the three examples show that the present method is more advantageous than TRKM.

scholarly journals On approximate solutions of a certain hyperbolic partial differential equation

2011 ◽  
Vol 42 (1) ◽  
pp. 95-104
Author(s):  
B. G. Pachpatte
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Integral Inequality ◽  
Initial Boundary ◽  
Approximate Solutions ◽  
Hyperbolic Partial Differential Equation ◽  
Explicit Estimate ◽  
Partial Differential ◽  
Discrete Analogues

In this paper we study approximatesolutions of a certain hyperbolic partial differential equation withthe given initial boundary conditions. A variant of a certainfundamental integral inequality with explicit estimate is used toestablish the results. The discrete analogues of the main resultsare also given.

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