scholarly journals A hybrid boundary integral/Taylor series approach to some nonlinear equations from fluid mechanics

1987 ◽  
Vol 28 (3) ◽  
pp. 279-286 ◽  
Author(s):  
C. J. Coleman
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Nonlinear Equations ◽  
Taylor Series ◽  
Nonlinear Partial Differential Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Boundary Integral Methods ◽  
Integral Methods

AbstractWe show that a combination of Taylor series and boundary integral methods can lead to an effective scheme for solving a class of nonlinear partial differential equations. The method is illustrated through its application to an equation from two dimensional fluid mechanics.

Error analysis of boundary integral methods

Acta Numerica ◽  
1992 ◽  
Vol 1 ◽  
pp. 287-339 ◽  
Author(s):  
Ian H. Sloan
Keyword(s):  
Approximate Solution ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Error Analysis ◽  
Boundary Value Problems ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Boundary Integral Methods ◽  
Integral Methods

Many of the boundary value problems traditionally cast as partial differential equations can be reformulated as integral equations over the boundary. After an introduction to boundary integral equations, this review describes some of the methods which have been proposed for their approximate solution. It discusses, as simply as possible, some of the techniques used in their error analysis, and points to areas in which the theory is still unsatisfactory.

scholarly journals Quantum Algorithms for Nonlinear Equations in Fluid Mechanics

2020 ◽  
Author(s):  
Rene Steijl
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Nonlinear Equations ◽  
Quantum Algorithms ◽  
Nonlinear Partial Differential Equations ◽  
Floating Point ◽  
Partial Differential ◽  
Linear Ordinary Differential Equations ◽  
Near Term

In recent years, significant progress has been made in the development of quantum algorithms for linear ordinary differential equations as well as linear partial differential equations. There has not been similar progress in the development of quantum algorithms for nonlinear differential equations. In the present work, the focus is on nonlinear partial differential equations arising as governing equations in fluid mechanics. First, the key challenges related to nonlinear equations in the context of quantum computing are discussed. Then, as the main contribution of this work, quantum circuits are presented that represent the nonlinear convection terms in the Navier–Stokes equations. The quantum algorithms introduced use encoding in the computational basis, and employ arithmetic based on the Quantum Fourier Transform. Furthermore, a floating-point type data representation is used instead of the fixed-point representation typically employed in quantum algorithms. A complexity analysis shows that even with the limited number of qubits available on current and near-term quantum computers (<100), nonlinear product terms can be computed with good accuracy. The importance of including sub-normal numbers in the floating-point quantum arithmetic is demonstrated for a representative example problem. Further development steps required to embed the introduced algorithms into larger-scale algorithms are discussed.

Two-dimensional bifurcation phenomena in thermal convection in horizontal, concentric annuli containing saturated porous media

1988 ◽  
Vol 187 ◽  
pp. 267-300 ◽  
Author(s):  
K. Himasekhar ◽  
Haim H. Bau
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Real Axis ◽  
Thermal Convection ◽  
Saturated Porous Medium ◽  
Nonlinear Partial Differential Equations ◽  
Perturbation Expansion ◽  
Two Dimensional ◽  
Partial Differential ◽  
The Real

A saturated porous medium confined between two horizontal cylinders is considered. As a result of a temperature difference between the cylinders, thermal convection is induced in the medium. The flow structure is investigated in a parameter space (R, Ra) where R is the radii ratio and Ra is the Darcy-Rayleigh number. In particular, the cases of R = 2, 2½, 21/4 and 2½ are considered. The fluid motion is described by the two-dimensional Darcy-Oberbeck-Boussinesq's (DOB) equations, which we solve using regular perturbation expansion. Terms up to O(Ra60) are calculated to obtain a series presentation for the Nusselt number Nu in the form \[ Nu(Ra^2) = \sum_{s=0}^{30} N_sRa^{2s}. \] This series has a limited range of utility due to singularities of the function Nu(Ra). The singularities lie both on and off the real axis in the complex Ra plane. For R = 2, the nearest singularity lies off the real axis, has no physical significance, and unnecessarily limits the range of utility of the aforementioned series. For R = 2½, 2¼ and 21/8, the singularity nearest to the origin is real and indicates that the function Nu(Ra) is no longer unique beyond the singular point.Depending on the radii ratio, the loss of uniqueness may occur as a result of either (perfect) bifurcations or the appearance of isolated solutions (imperfect bifurcations). The structure of the multiple solutions is investigated by solving the DOB equations numerically. The nonlinear partial differential equations are converted into a truncated set of ordinary differential equations via projection. The steady-state problem is solved using Newton's technique. At each step the determinant of the Jacobian is evaluated. Bifurcation points are identified with singularities of the Jacobian. Linear stability analysis is used to determine the stability of various solution branches. The results we obtained from solving the DOB equations using perturbation expansion are compared with those we obtained from solving the nonlinear partial differential equations numerically and are found to agree well.

scholarly journals Nonlinear Klein-Gordon and Schrödinger Equations by the Projected Differential Transform Method

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
Younghae Do ◽  
Bongsoo Jang
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Taylor Series ◽  
Nonlinear Partial Differential Equations ◽  
Differential Transform Method ◽  
Schrödinger Equations ◽  
Partial Differential ◽  
Transform Method ◽  
Differential Transform ◽  
Klein Gordon

The differential transform method (DTM) is based on the Taylor series for all variables, but it differs from the traditional Taylor series in calculating coefficients. Even if the DTM is an effective numerical method for solving many nonlinear partial differential equations, there are also some difficulties due to the complex nonlinearity. To overcome difficulties arising in DTM, we present the new modified version of DTM, namely, the projected differential transform method (PDTM), for solving nonlinear partial differential equations. The proposed method is applied to solve the various nonlinear Klein-Gordon and Schrödinger equations. Numerical approximations performed by the PDTM are presented and compared with the results obtained by other numerical methods. The results reveal that PDTM is a simple and effective numerical algorithm.

B-determining equations: applications to nonlinear partial differential equations

1995 ◽  
Vol 6 (3) ◽  
pp. 265-286 ◽  
Author(s):  
O. V. Kaptsov
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Fluid Mechanics ◽  
Exact Solutions ◽  
Nonlinear Partial Differential Equations ◽  
Symmetry Groups ◽  
Partial Differential ◽  
Physical Importance

We introduce the concept of B-determining equations of a system of partial differential equations that generalize the defining equations of the symmetry groups. We show how this concept may be applied to obtain exact solutions of partial differential equations. The exposition is reasonable self-contained, and supplemented by examples of direct physical importance, chosen from fluid mechanics.

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