Ordinary and Partial Differential Equations with Initial and Boundary Values

We avoid as as much as possible the order reduction of Rosenbrock methods when they are applied to nonlinear partial differential equations by means of a similar technique to the one used previously by us for the linear case. For this we use a suitable choice of boundary values for the internal stages. The main difference from the linear case comes from the difficulty to calculate those boundary values exactly in terms of data. In any case, the implementation is cheap and simple since, at each stage, just some additional terms concerning those boundary values and not the whole grid must be added to what would be the standard method of lines.

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A numerical technique for linear elliptic partial differential equations in polygonal domains

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2014.0747 ◽

2015 ◽

Vol 471 (2175) ◽

pp. 20140747 ◽

Cited By ~ 14

Author(s):

P. Hashemzadeh ◽

A. S. Fokas ◽

S. A. Smitheman

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Elliptic Partial Differential Equations ◽

Boundary Values ◽

Physical Plane ◽

Partial Differential ◽

Fourier Plane ◽

Polygonal Domains ◽

Collocation Points ◽

Global Relations

Integral representations for the solution of linear elliptic partial differential equations (PDEs) can be obtained using Green's theorem. However, these representations involve both the Dirichlet and the Neumann values on the boundary, and for a well-posed boundary-value problem (BVPs) one of these functions is unknown. A new transform method for solving BVPs for linear and integrable nonlinear PDEs usually referred to as the unified transform ( or the Fokas transform ) was introduced by the second author in the late Nineties. For linear elliptic PDEs, this method can be considered as the analogue of Green's function approach but now it is formulated in the complex Fourier plane instead of the physical plane. It employs two global relations also formulated in the Fourier plane which couple the Dirichlet and the Neumann boundary values. These relations can be used to characterize the unknown boundary values in terms of the given boundary data, yielding an elegant approach for determining the Dirichlet to Neumann map . The numerical implementation of the unified transform can be considered as the counterpart in the Fourier plane of the well-known boundary integral method which is formulated in the physical plane. For this implementation, one must choose (i) a suitable basis for expanding the unknown functions and (ii) an appropriate set of complex values, which we refer to as collocation points, at which to evaluate the global relations. Here, by employing a variety of examples we present simple guidelines of how the above choices can be made. Furthermore, we provide concrete rules for choosing the collocation points so that the condition number of the matrix of the associated linear system remains low.

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Parabolic Partial Differential Equations with Nonlocal Initial and Boundary Values

International Journal of Computational Methods ◽

10.1142/s0219876215500243 ◽

2015 ◽

Vol 12 (05) ◽

pp. 1550024 ◽

Cited By ~ 8

Author(s):

M. Turkyilmazoglu

Keyword(s):

Partial Differential Equations ◽

Differential Equations ◽

Series Representation ◽

Real Life ◽

Parabolic Partial Differential Equation ◽

Parabolic Partial Differential Equations ◽

Algebraic Equations ◽

Boundary Values ◽

Partial Differential ◽

Double Power Series

Parabolic partial differential equations possessing nonlocal initial and boundary specifications are used to model some real-life applications. This paper focuses on constructing fast and accurate analytic approximations via an easy, elegant and powerful algorithm based on a double power series representation of the solution via ordinary polynomials. Consequently, a parabolic partial differential equation is reduced to a system involving algebraic equations. Exact solutions are obtained when the solutions are themselves polynomials. Some parabolic partial differential equations are treated by the technique to judge its validity and also to measure its accuracy as compared to the existing methods.

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