Catalogue Search | MBRL
Are you sure you want to remove the book from the shelf?
{{itemTitle}}
1,280
result(s) for
"Separation of variables."
Sort by:
Separation of variables for partial differential equations : an eigenfunction approach
\"Written at the advanced undergraduate level, the book will serve equally well as a text for students and as a reference for instructors and users of separation of variables. It requires a background in engineering mathematics, but no prior exposure to separation of variables. The abundant worked examples provide guidance for deciding whether and how to apply the method to any given problem, help in interpreting computed solutions, and give insight into cases in which formal answers may be useless\"--Jacket.
Book
Functional Separation of Variables in Nonlinear PDEs: General Approach, New Solutions of Diffusion-Type Equations
The study gives a brief overview of existing modifications of the method of functional separation of variables for nonlinear PDEs. It proposes a more general approach to the construction of exact solutions to nonlinear equations of applied mathematics and mathematical physics, based on a special transformation with an integral term and the generalized splitting principle. The effectiveness of this approach is illustrated by nonlinear diffusion-type equations that contain reaction and convective terms with variable coefficients. The focus is on equations of a fairly general form that depend on one, two or three arbitrary functions (such nonlinear PDEs are most difficult to analyze and find exact solutions). A lot of new functional separable solutions and generalized traveling wave solutions are described (more than 30 exact solutions have been presented in total). It is shown that the method of functional separation of variables can, in certain cases, be more effective than (i) the nonclassical method of symmetry reductions based on an invariant surface condition, and (ii) the method of differential constraints based on a single differential constraint. The exact solutions obtained can be used to test various numerical and approximate analytical methods of mathematical physics and mechanics.
Journal Article
Symmetric Separation of Variables for the Extended Clebsch and Manakov Models
In the present paper, using a modification of the method of vector fields$Z_i$of the bi-Hamiltonian theory of separation of variables (SoV), we construct symmetric non-Stäckel variable separation for three-dimensional extension of the Clebsch model, which is equivalent (in the bi-Hamiltonian sense) to the system of interacting Manakov (Schottky-Frahm) and Euler tops. For the obtained symmetric SoV (contrary to the previously constructed asymmetric one), all curves of separation are the same and have genus five. It occurred that the difference between the symmetric and asymmetric cases is encoded in the different form of the vector fields$Z$used to construct separating polynomial. We explicitly construct coordinates and momenta of separation and Abel-type equations in the considered examples of symmetric SoV for the extended Clebsch and Manakov models.
Journal Article
Computational Methods for Solving Higher-Order (1+1) Dimensional Mixed-Difference Integro-Differential Equations with Variable Coefficients
The main purpose of this article is to present a new technique for solving (1+1) mixeddimensional difference integro-differential Equations (2D-MDeIDEs) in position and time with coefficients of variables under mixed conditions. The equations proposed for the solution represent a link between time and delay in position that has not been previously studied. Therefore, the authors used the technique of separation of variables to transform the 2D-MDeIDE into one-dimensional Fredholm difference integro-differential Equations (FDeIDEs), and then using the Bernoulli polynomial method (BPM), we obtained a system of linear algebraic equations (SLAE). The other aspect of the technique of separation of variables is explicitly obtaining the necessary and appropriate time function to obtain the best numerical results. Some numerical experiments are performed to show the simplicity and efficiency of the presented method, and all results are performed by Maple 18.
Journal Article
Numerical simulation, existence and uniqueness for solving nonlinear mixed partial integro-differential equations with discontinuous kernels
This study describes a new effective technique for solving mixed partial integro-differential equations that are nonlinear with discontinuous kernels (NMPI-DEs). We have used two well-known different numerical techniques, the toeplitz matrix technique (TMT), and the product Nystrom technique (PNT). We have outlined the characteristics of TMT and PNT in both cases, as well as the significance of each approach for characterizing and demystifying the problems’ complexity. These methods have used to convert a system of nonlinear algebraic equations has been derived from the nonlinear Fredholm integral equation (NFIE). Banach’s fixed point theory is employed to investigate the existence and uniqueness of the solution to the nonlinear mixed integral problem. Compared to other approaches, these strategies have shown excellent results in the first instance of being utilized to solve this kind of complex problem. Lastly, a comparison of the two distinct approaches is shown using several cases by using tables and figures. The Maple software has been utilized to compute and obtain all of the numerical results.
Journal Article
Fourier, Gegenbauer and Jacobi Expansions for a Power-Law Fundamental Solution of the Polyharmonic Equation and Polyspherical Addition Theorems
We develop complex Jacobi, Gegenbauer and Chebyshev polynomial expansions for the kernels associated with power-law fundamental solutions of the polyharmonic equation on d-dimensional Euclidean space. From these series representations we derive Fourier expansions in certain rotationally-invariant coordinate systems and Gegenbauer polynomial expansions in Vilenkin's polyspherical coordinates. We compare both of these expansions to generate addition theorems for the azimuthal Fourier coefficients. [ProQuest: [...] denotes formulae omitted.]
Journal Article
Symmetries and Separation of Variables
In this paper, we look at the method of separation of variables of a PDE from its symmetry transformation point of view. Specifically, we discuss the relation between the existence of additively and multiplicatively separated variables of a PDE, and the form of its symmetry operators. We show that solutions in the form of separated variables are in fact, invariant solutions, i.e. solutions invariant under some subalgebra of the symmetry operators of the equation. For the case of two independent variables, we obtain the form of Lie point symmetry operators corresponding to additively and multiplicatively separated solutions, and generalize our results for the case when separated variables are any functions of independent variables. We also discuss the role of contact symmetry transformations and differential invariants for the existence of separated solutions, and outline the role of variational symmetries, as well as conditional (non-classical) symmetry operators. We demonstrate that the symmetry approach is a valuable tool for obtaining information regarding existence of solutions with separated variables.
Journal Article
Numerical solution and dynamical studies for (2 + 1) dimensional Volterra–Fredholm integral equations with a discontinuous kernel
In this paper, using the space
L
2
(
[
a
,
b
]
×
[
c
,
d
]
)
×
C
[
0
,
T
]
,
(
T
<
1
)
, we have introduced a new and efficient approach to a discontinuous kernel solution of (2 + 1) dimensional mixed Volterra–Fredholm integral equations (
MVFIE
). By applying the separation of variables approach, the (
2 + 1
) dimensional
MVFIE
has been reduced to a two-dimensional Fredholm integral Eq. (
2D-FIE
). Next, we derive a system of linear algebraic equations using the Chebyshev polynomials of the sixth-kind (
CP6K
) approach. We are demonstrating that the integral equations solution exists and unique. Convergence of solutions has been proven. Numerical simulations have been given to verify that the new method produces more efficient and better results. The error in each example is computed by using Maple software.
Journal Article
Raising Two More Fundamental Questions Regarding the Classical Views on the Rheology of Polymer Melts
The current paradigm of polymer flow assumes that (i) the effect of the molecular weight of the macromolecules, M, and of the temperature, T, on the expression of the viscosity of polymer melts separate; (ii) the molecular weight for entanglement, Mc, is independent of T; and (iii) the determination of Mc by the break in the log viscosity curve against log M unequivocally differentiates un-entangled melts from entangled melts. We use reliable rheological data on monodispersed polystyrene samples from very low molecular weight (M/Mc = 0.015) to relatively high molecular weight (M/Mc = 34) to test the separation of M and T in the expression of the viscosity; we reveal that an overall illusion of the validity of the separation of T and M is mathematically comprehensible, especially at high temperature and for M > 2Mc, but that, strictly speaking, the separation of M and T is not valid, except for certain periodic values of M equal to Mc, 2Mc, 4Mc, 8Mc, 16Mc, etc. (period doubling) organized around a “pole reference” value MR = 4Mc. We also reveal, for M < Mc, the existence of a lower molecular weight limit, M’c = Mc/8 for the onset of the macromolecular behavior (macro-coil). The discrete and periodic values of M that validate the separation of the effect of M and T on the viscosity generate the fragmentation of the molecular range into three rheological ranges. Likewise, we show that the effect of temperature is also fragmented into three rheological ranges for T > Tg: Tg < T< (Tg + 23°), (Tg + 23°) < T < TLL and T > TLL’ where TLL is the liquid-liquid temperature. Our conclusion is that the classical formulation of the viscosity of polymer melts is so overly simplified that it is missing important experimental facts, such as period doubling for the separation of T and M, TLL, M’c, and Mc, resulting in its inability to understand the true nature of entanglements. We present in the discussion of the paper the alternative approach to the viscoelastic behavior, “the duality and cross-duality” of the Dual-conformers, showing how this model formalism was used to test mathematically and invalidate the separation of T and M in the classical formulation of viscosity.
Journal Article
Multidimensional Nonautonomous Evolution Monge–Ampère Type Equations
We study multidimensional nonautonomous evolution Monge–Ampère type equations. The left-hand side of such equation contains the first time derivative with the coefficient depending on time, space variables, and an unknown function. The right-hand side of the equation is the determinant of a Hessian matrix. We find the solutions by additive and multiplicative separation of variables and show that the representability of the coefficient of the time derivative as the product of functions of time and space variables is a sufficient condition for the existence of such solutions. In the case that the coefficient of the time derivative is the inverse function to a linear combination of space variables with coefficients depending on time, we also give solutions in the form of the quadratic polynomials in space variables. Also, we obtain the solution set in the form of the linear combination of functions of space variables with time depending coefficients. We consider some reductions of the equation to ODEs in the cases that the unknown function depends on the sum of functions of space variables (in particular, the sum of their squares) and a function of the time; in this case we use the functional separation of variables. Some reductions are also found of the given equation to PDEs of lower dimension. In particular, we find the solutions in the form of function of the time and the sum of squares of space variables as well as the solutions in the form of the sum of such functions.
Journal Article