scholarly journals Integrability analysis of the partial differential equation describing the classical bond-pricing model of mathematical finance

Open Physics ◽  
2018 ◽  
Vol 16 (1) ◽  
pp. 766-779
Author(s):  
Taha Aziz ◽  
Aeeman Fatima ◽  
Chaudry Masood Khalique
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Equation ◽  
Canonical Form ◽  
Mathematical Finance ◽  
Fundamental Solutions ◽  
Bond Pricing ◽  
Canonical Forms ◽  
Pricing Model ◽  
Invariant Approach

AbstractThe invariant approach is employed to solve the Cauchy problem for the bond-pricing partial differential equation (PDE) of mathematical finance. We first briefly review the invariant criteria for a scalar second-order parabolic PDE in two independent variables and then utilize it to reduce the bond-pricing equation to different Lie canonical forms. We show that the invariant approach aids in transforming the bond-pricing equation to the second Lie canonical form and that with a proper parametric selection, the bond-pricing PDE can be converted to the first Lie canonical form which is the classical heat equation. Different cases are deduced for which the original equation reduces to the first and second Lie canonical forms. For each of the cases, we work out the transformations which map the bond-pricing equation into the heat equation and also to the second Lie canonical form. We construct the fundamental solutions for the bond-pricing model via these transformations by utilizing the fundamental solutions of the classical heat equation as well as solution to the second Lie canonical form. Finally, the closed-form analytical solutions of the Cauchy initial value problems for the bond-pricing model with proper choice of terminal conditions are obtained.

scholarly journals On the Resolution of a Remarkable Bond Pricing Model from Financial Mathematics: Application of the Deductive Group Theoretical Technique

2021 ◽  
Vol 2021 ◽  
pp. 1-10
Author(s):  
Taha Aziz
Keyword(s):  
Heat Equation ◽  
Linear Space ◽  
Canonical Form ◽  
Space Time ◽  
Time Dependent ◽  
Canonical Forms ◽  
Pricing Model ◽  
Partial Differential ◽  
Equivalence Transformations ◽  
Parabolic Pde

The classical Cox–Ingersoll–Ross (CIR) bond-pricing model is based on the evolution space-time dependent partial differential equation (PDE) which represents the standard European interest rate derivatives. In general, such class of evolution partial differential equations (PDEs) has generally been resolved by classical methods of PDEs and by ansatz-based techniques which have been previously applied in a similar context. The author here shows the application of an invariant approach, a systematic method based on deductive group-theoretical analysis. The invariant technique reduces the scalar linear space-time dependent parabolic PDE to one of the four classical Lie canonical forms. This method leads us to exactly solve the scalar linear space-time dependent parabolic PDE representing the CIR model. It was found that CIR PDE is transformed into the first canonical form, which is the heat equation. Under the proper choice of emerging parameters of the model, the CIR equation is also reduced to the second Lie canonical form. The equivalence transformations which map the CIR PDE into the different canonical forms are deduced. With the use of these equivalence transformations, the invariant solutions of the underlying model are found by using some well-known results of the heat equation and the second Lie canonical form. Furthermore, the Cauchy initial-value model of the CIR problem along with the terminal condition is discussed and closed-form solutions are deduced. Finally, the conservation laws associated with the CIR equation are derived by using the general conservation theorem.

scholarly journals Analytic transient solutions of a cylindrical heat equation

Filomat ◽  
2021 ◽  
Vol 35 (8) ◽  
pp. 2617-2628
Author(s):  
K.Y. Kung ◽  
Man-Feng Gong ◽  
H.M. Srivastava ◽  
Shy-Der Lin
Keyword(s):  
Heat Transfer ◽  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Heat Equation ◽  
Heat Conduction Equation ◽  
Separation Of Variables ◽  
Transient Heat Conduction ◽  
Transient Temperature ◽  
Transient Solutions

The principles of superposition and separation of variables are used here in order to investigate the analytical solutions of a certain transient heat conduction equation. The structure of the transient temperature appropriations and the heat-transfer distributions are summed up for a straight mix of the results by means of the Fourier-Bessel arrangement of the exponential type for the investigated partial differential equation.

Polynomial approximations to the solution of the heat equation

1973 ◽  
Vol 73 (1) ◽  
pp. 157-165 ◽  
Author(s):  
R. E. Scraton
Keyword(s):  
Differential Equation ◽  
Boundary Condition ◽  
Partial Differential Equation ◽  
Exact Solution ◽  
Heat Equation ◽  
Least Squares ◽  
Linear Boundary ◽  
Chebyshev Series ◽  
Linear Boundary Condition ◽  
Image Position

AbstractAn approximation is found to the solution of the partial differential equationin the region −1 ≤ x ≤ 1, t > 0, where u satisfies a general linear boundary condition on x = ± 1. This approximation is a polynomial in x, and is an exact solution of a perturbed form of the differential equation. By choosing the perturbation appropriately, this approach is mathematically equivalent to some recent methods for solving the differential equation in the form of a Chebyshev series. Better approximations to the required solution (and particularly to the eigenvalues) are obtained by choosing the perturbation to satisfy a least squares criterion.

Global solution of nonlinear stochastic heat equation with solutions in a Hilbert manifold

2020 ◽  
Vol 20 (06) ◽  
pp. 2040012
Author(s):  
Zdzisław Brzeźniak ◽  
Javed Hussain
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Hilbert Space ◽  
Evolution Equation ◽  
Heat Equation ◽  
Global Solution ◽  
Unit Sphere ◽  
Stochastic Partial Differential Equation ◽  
Hilbert Manifold ◽  
Unconstrained Problem

The objective of this paper is to prove the existence of a global solution to a certain stochastic partial differential equation subject to the [Formula: see text]-norm being constrained. The corresponding evolution equation can be seen as the projection of the unconstrained problem onto the tangent space of the unit sphere [Formula: see text] in a Hilbert space [Formula: see text].

scholarly journals Explicit Solution for Cylindrical Heat Conduction

2016 ◽  
Vol 13 (2) ◽  
Author(s):  
Kaitlyn Parsons ◽  
Tyler Reichanadter ◽  
Andi Vicksman ◽  
Harvey Segur
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Heat Conduction ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Analytical Method ◽  
Explicit Solution ◽  
Separation Of Variables ◽  
Partial Differential ◽  
Forcing Function

The heat equation is a partial differential equation that elegantly describes heat conduction or other diffusive processes. Primary methods for solving this equation require time-independent boundary conditions. In reality this assumption rarely has any validity. Therefore it is necessary to construct an analytical method by which to handle the heat equation with time-variant boundary conditions. This paper analyzes a physical system in which a solid brass cylinder experiences heat flow from the central axis to a heat sink along its outer rim. In particular, the partial differential equation is transformed such that its boundary conditions are zero which creates a forcing function in the transform PDE. This transformation constructs a Green’s function, which admits the use of variation of parameters to find the explicit solution. Experimental results verify the success of this analytical method. KEYWORDS: Heat Equation; Bessel-Fourier Decomposition; Cylindrical; Time-dependent Boundary Conditions; Orthogonality; Partial Differential Equation; Separation of Variables; Green’s Functions

scholarly journals Solving Fundamental Solution of Non-Homogeneous Heat Equation with Dirichlet Boundary Conditions

2020 ◽  
Vol 22 ◽  
pp. 1-9
Author(s):  
Kahsay Godifey Wubneh
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Boundary Conditions ◽  
Heat Equation ◽  
Constant Coefficient ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Partial Differential ◽  
Nonhomogeneous Heat ◽  
Interesting Solution

In this study, we developed a solution of nonhomogeneous heat equation with Dirichlet boundary conditions. moreover, the non-homogeneous heat equation with constant coefficient. since heat equation has a simple form, we would like to start from the heat equation to find the exact solution of the partial differential equation with constant coefficient. to emphasize our main results, we also consider some important way of solving of partial differential equation specially solving heat equation with Dirichlet boundary conditions. the main results of our paper are quite general in nature and yield some interesting solution of non-homogeneous heat equation with Dirichlet boundary conditions and it is used for problems of mathematical modeling and mathematical physics.

Influence of Some Parameters on the Solution of a Thermal Model of Vulcanization

1993 ◽  
Vol 66 (1) ◽  
pp. 19-29 ◽  
Author(s):  
Jacques Burger ◽  
Nicole Burger ◽  
Marc Pogu
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Finite Element ◽  
Heat Equation ◽  
Thermal Behavior ◽  
Thermal Model ◽  
Nonlinear Heat Equation ◽  
Reaction Equation ◽  
Finite Element Procedure ◽  
Vulcanization Process

Abstract The thermal behavior of a piece of rubber during a vulcanization process is modelized by a set of two coupled differential equations. The first one is a classical nonlinear heat equation while the second is a reaction equation. The aim of this article is first to solve numerically the two equations using a finite element procedure for the partial differential equation and a Runge-Kutta like scheme for the reaction equation, and then to study the sensitivity of the solution to variations in some parameters.

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