Similarity Solutions of Partial Differential Equations in Probability

Two-dimensional diffusion processes are considered between concentric circles and in angular sectors. The aim of the paper is to compute the probability that the process will hit a given part of the boundary of the stopping region first. The appropriate partial differential equations are solved explicitly by using the method of similarity solutions and the method of separation of variables. Some solutions are expressed as generalized Fourier series.

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Separation of Variables for Systems of First-Order Partial Differential Equations and the Dirac Equation in Two-Dimensional Manifolds

Symmetries and Overdetermined Systems of Partial Differential Equations - The IMA Volumes in Mathematics and its Applications ◽

10.1007/978-0-387-73831-4_25 ◽

2008 ◽

pp. 471-496 ◽

Cited By ~ 2

Author(s):

Raymond G. Mclenaghan ◽

Giovanni Rastelli

Keyword(s):

Partial Differential Equations ◽

Dirac Equation ◽

Differential Equations ◽

Separation Of Variables ◽

Two Dimensional ◽

Partial Differential ◽

First Order

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Itô's Stochastic Partial Differential Equations as Non-Markov Models Leading to High-Dimensional Diffusion Processes

High-Dimensional Nonlinear Diffusion Stochastic Processes - Series on Advances in Mathematics for Applied Sciences ◽

10.1142/9789812810540_0005 ◽

2001 ◽

pp. 141-161

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Stochastic Partial Differential Equations ◽

Markov Models ◽

Diffusion Processes ◽

High Dimensional ◽

Partial Differential ◽

Dimensional Diffusion

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Transformation of elliptic partial differential equations for solving two-dimensional boundary value problems in fluid flow

International Journal for Numerical Methods in Engineering ◽

10.1002/nme.1620100304 ◽

1976 ◽

Vol 10 (3) ◽

pp. 527-533 ◽

Cited By ~ 13

Author(s):

J. D. Boadway

Keyword(s):

Fluid Flow ◽

Partial Differential Equations ◽

Differential Equations ◽

Boundary Value Problems ◽

Boundary Value ◽

Elliptic Partial Differential Equations ◽

Two Dimensional ◽

Partial Differential ◽

Dimensional Boundary

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Numerical approximation of linear operator equations using a generalized Fourier series: ordinary and partial differential equations with boundary conditions

Applied Mathematical Modelling ◽

10.1016/0307-904x(89)90207-2 ◽

1989 ◽

Vol 13 (10) ◽

pp. 601-614 ◽

Cited By ~ 2

Author(s):

Theodore V. Hromadka ◽

Robert J. Whitley

Keyword(s):

Fourier Series ◽

Partial Differential Equations ◽

Linear Operator ◽

Differential Equations ◽

Boundary Conditions ◽

Numerical Approximation ◽

Operator Equations ◽

Partial Differential ◽

Generalized Fourier Series ◽

Linear Operator Equations

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Beam Vibrations With Time-Dependent Boundary Conditions

Journal of Applied Mechanics ◽

10.1115/1.4010161 ◽

1950 ◽

Vol 17 (4) ◽

pp. 377-380

Author(s):

R. D. Mindlin ◽

L. E. Goodman

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Boundary Conditions ◽

Boundary Value Problems ◽

Separation Of Variables ◽

Boundary Value ◽

Time Dependent ◽

Partial Differential ◽

Linear Partial Differential Equations ◽

Vibration Problems

Abstract A procedure is described for extending the method of separation of variables to the solution of beam-vibration problems with time-dependent boundary conditions. The procedure is applicable to a wide variety of time-dependent boundary-value problems in systems governed by linear partial differential equations.

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An efficient Bi-cubic B-spline ADI method for numerical solutions of two-dimensional unsteady advection diffusion equations

International Journal of Numerical Methods for Heat &amp Fluid Flow ◽

10.1108/hff-12-2017-0511 ◽

2018 ◽

Vol 28 (11) ◽

pp. 2620-2649 ◽

Cited By ~ 2

Author(s):

Rajni Rohila ◽

R.C. Mittal

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Numerical Solutions ◽

Diffusion Equations ◽

Spline Functions ◽

Two Dimensional ◽

Partial Differential ◽

Content Type ◽

B Spline ◽

Advection Diffusion

Purpose This paper aims to develop a novel numerical method based on bi-cubic B-spline functions and alternating direction (ADI) scheme to study numerical solutions of advection diffusion equation. The method captures important properties in the advection of fluids very efficiently. C.P.U. time has been shown to be very less as compared with other numerical schemes. Problems of great practical importance have been simulated through the proposed numerical scheme to test the efficiency and applicability of method. Design/methodology/approach A bi-cubic B-spline ADI method has been proposed to capture many complex properties in the advection of fluids. Findings Bi-cubic B-spline ADI technique to investigate numerical solutions of partial differential equations has been studied. Presented numerical procedure has been applied to important two-dimensional advection diffusion equations. Computed results are efficient and reliable, have been depicted by graphs and several contour forms and confirm the accuracy of the applied technique. Stability analysis has been performed by von Neumann method and the proposed method is shown to satisfy stability criteria unconditionally. In future, the authors aim to extend this study by applying more complex partial differential equations. Though the structure of the method seems to be little complex, the method has the advantage of using small processing time. Consequently, the method may be used to find solutions at higher time levels also. Originality/value ADI technique has never been applied with bi-cubic B-spline functions for numerical solutions of partial differential equations.

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SOLVING SYSTEMS OF FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS VIA TWO DIMENSIONAL LAPLACE TRANSFORMS

International Journal of Research -GRANTHAALAYAH ◽

10.29121/granthaalayah.v5.i12.2017.528 ◽

2020 ◽

Vol 5 (12) ◽

pp. 406-420

Author(s):

A. Aghili ◽

M.R. Masomi

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Laplace Transform ◽

Laplace Transforms ◽

Heat Equations ◽

Two Dimensional ◽

Fractional Partial Differential Equations ◽

Partial Differential ◽

Homogeneous Systems ◽

Fractional Pde

In this article, the authors used two dimensional Laplace transform to solve non - homogeneous sub - ballistic fractional PDE and homogeneous systems of time fractional heat equations. Constructive examples are also provided.

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