scholarly journals A new transform method II: the global relation and boundary-value problems in polar coordinates

2010 ◽  
Vol 466 (2120) ◽  
pp. 2283-2307 ◽  
Author(s):  
E. A. Spence ◽  
A. S. Fokas
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Integral Representations ◽  
New Method ◽  
Polar Coordinates ◽  
Nonlinear Pdes ◽  
Boundary Values ◽  
Transform Method ◽  
Global Relation ◽  
The Fourier Transform

A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel integral representations (IRs) that are formulated in the Fourier (transform) space. In a previous paper, a simplified way of obtaining these representations was presented. In the current paper, first, the second ingredient of the new method, namely the derivation of the so-called ‘global relation’ (GR)—an equation involving transforms of the boundary values—is presented. Then, using the GR as well as the IR derived in the previous paper, certain BVPs in polar coordinates are solved. These BVPs elucidate the fact that this method has substantial advantages over the classical transform method.

scholarly journals A new transform method I: domain-dependent fundamental solutions and integral representations

2010 ◽  
Vol 466 (2120) ◽  
pp. 2259-2281 ◽  
Author(s):  
E. A. Spence ◽  
A. S. Fokas
Keyword(s):  
Fourier Transform ◽  
Fundamental Solutions ◽  
Integral Representations ◽  
New Method ◽  
Polar Coordinates ◽  
Nonlinear Pdes ◽  
Elliptic Pdes ◽  
Transform Method ◽  
The Fourier Transform ◽  
Linear Pdes

A new method for solving boundary-value problems (BVPs) for linear and certain nonlinear PDEs was introduced by one of the authors in the late 1990s. For linear PDEs, this method constructs novel integral representations (IRs) that are formulated in the Fourier (transform) space. In this paper, we present a simplified way of obtaining these representations for elliptic PDEs; namely, we introduce an algorithm for constructing particular, domain-dependent, IRs of the associated fundamental solutions, which are then substituted into Green's IRs. Furthermore, we extend this new method from BVPs in polygons to BVPs in polar coordinates. In the sequel to this paper, these results are used to solve particular BVPs, which elucidate the fact that this method has substantial advantages over the classical transform method.

scholarly journals Integral formulation of solutions of boundary-value problems of heat and mass transfer in domains with moving boundaries

2021 ◽  
Vol 29 (1) ◽  
pp. 73-91
Author(s):  
Valentin V. Shevelev
Keyword(s):  
Heat Conduction ◽  
Boundary Value Problems ◽  
Moving Boundary ◽  
Boundary Value ◽  
Phase Region ◽  
Integral Representations ◽  
Two Phase ◽  
Self Similar ◽  
The Fourier Transform ◽  
And Diffusion

Using the Fourier transform, integral representations of solutions to boundary value problems of heat conduction and diffusion in a two-phase region with a moving interface are obtained. The proposed approach makes it possible to obtain the equation of motion of the interface without the need to first find the temperature and (or) concentration fields. This makes it possible to study the stability of the interface with respect to disturbances in its shape. The validity of the proposed approach is demonstrated by the example of self-similar growth of a spherical crystal in a supercooled melt and crystallization of the melt on a substrate of the same substance. On the basis of the obtained equation, which determines the rate of self-similar motion of the interface, the features of the kinetics of crystallization of the melt on the substrate are analyzed. The conditions of applicability of the developed approach to the solution of boundary value problems of heat conduction and diffusion in regions separated by a moving boundary are briefly discussed.

scholarly journals Solving Wiener–Hopf problems without kernel factorization

2014 ◽  
Vol 470 (2170) ◽  
pp. 20140304 ◽  
Author(s):  
Darren G. Crowdy ◽  
Elena Luca
Keyword(s):  
Fluid Mechanics ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Kernel Functions ◽  
New Method ◽  
Transform Method ◽  
New Approach ◽  
Mathematical Ideas ◽  
Analytical Functions ◽  
Unified Transform Method

A new approach to solving problems of Wiener–Hopf type is expounded by showing its implementation in two concrete and typical examples from fluid mechanics. The new method adapts mathematical ideas underlying the so-called unified transform method due to A. S. Fokas and collaborators in recent years. The method has the key advantage of avoiding what is usually the most challenging part of the usual Wiener–Hopf approach: the factorization of kernel functions into sectionally analytical functions. Two example boundary value problems, involving both harmonic and biharmonic fields, are solved in detail. The approach leads to fast and accurate schemes for evaluation of the solutions.

scholarly journals On the solutions of some boundary value problems by using differential transformation method with convolution terms

Filomat ◽  
2012 ◽  
Vol 26 (5) ◽  
pp. 917-928 ◽  
Author(s):  
Adem Kılıçman ◽  
Ömer Altun
Keyword(s):  
Differential Equations ◽  
Boundary Value Problems ◽  
Boundary Value ◽  
Partial Differential Operator ◽  
Transformation Method ◽  
New Method ◽  
Differential Transformation Method ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transformation

In this study, we consider some boundary value problems by using the differential transformation method with convolutions term. Further, we also propose a new method to solve the differential equations having singularity by using the convolution. In this new method when the operator has some singularities then we multiply the partial differential operator with continuously differential functions by using the convolution in order to regularize the singularity. Then the differential transform method will be applied to the new partial differential equations that might also have some fractional order.

A new method for calculating eigenvalues with applications to gravity-capillary waves with edge constraints

1983 ◽  
Vol 94 (3) ◽  
pp. 553-564 ◽  
Author(s):  
James Graham-Eagle
Keyword(s):  
Boundary Value Problems ◽  
Variational Formulation ◽  
Boundary Value ◽  
Capillary Waves ◽  
Original Study ◽  
New Method ◽  
Linear Boundary ◽  
Hydrodynamic Problem ◽  
Alternative Means ◽  
Standard Linear

The method to be described provides an alternative means of dealing with certain non-standard linear boundary-value problems. It is developed in several applications to the theory of gravity-capillary waves. The analysis is based on a variational formulation of the hydrodynamic problem, being motivated by and extending the original study by Benjamin and Scott [3].

scholarly journals An axially symmetric forced convection problem

1975 ◽  
Vol 20 (1) ◽  
pp. 1-17
Author(s):  
J. A. Belward
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Convection Heat Transfer ◽  
Fundamental Solutions ◽  
Integral Representations ◽  
Original Equation ◽  
Transfer Model ◽  
Axially Symmetric ◽  
Simple Diffusion ◽  
Diffusion Convection

AbstractA simple diffusion-convection heat transfer model is formulated which leads to an axially symmetric partial differential equation. The equation is shown to be closely related to a second one which is adjoint to the original equation in one variable can and be interpreted as a description of another diffusion-convection model. Fundamental solutions of the original equation are constructed and interpreted with reference to both models. Some boundary value problems are solved in series form and integral representations of the solutions are also given. The boundary value problems are shown to be equivalent to an integral equation and the correspondence between the two formulations is understood in terms of the two diffusion-convection problems. A Péclet number is defined in one of the boundary value problems and the behaviour of the solutions is studied for large and small values of this parameter.

Sign in / Sign up

Export Citation Format

Share Document