227 research outputs found
Approximate Similarity Reduction
The nonlinear K (n;1) equation with damping is investigated via the approximate homotopy symmetry method and approximate homotopy direct method. The approximate homotopy symmetry and homotopy similarity reduction equations of different orders are derived and the corresponding homotopy series reduction solutionsare obtained. As a result, the formal coincidence for both methods is displayed
Approximate nonlinear self-adjointness and approximate conservation laws
In this paper, approximate nonlinear self-adjointness for perturbed PDEs is
introduced and its properties are studied. Consequently, approximate
conservation laws which cannot be obtained by the approximate Noether theorem
are constructed by means of the method. As an application, a class of perturbed
nonlinear wave equations is considered to illustrate the effectiveness.Comment: 13 pages, 2 table
Perturbation Iteration Transform Method for the Solution of Newell-Whitehead-Segel Model Equations
In this study, a computational method referred to as Perturbation
Iteration Transform Method (PITM), which is a combination of the
conventional Laplace Transform Method (LTM) and the Perturbation
Iteration Algorithm (PIA) is applied for the solution of Newell-Whitehead-
Segel Equations (NWSEs). Three unique examples are considered and the
results obtained are compared with their exact solutions graphically. Also,
the results agree with those obtained via other semi-analytical methods viz:
New Iterative Method and Adomian Decomposition Method. This present
method proves to be very efficient and reliable. Mathematica 10 is used for
all the computations in this stud
Towards computational Morse-Floer homology: forcing results for connecting orbits by computing relative indices of critical points
To make progress towards better computability of Morse-Floer homology, and
thus enhance the applicability of Floer theory, it is essential to have tools
to determine the relative index of equilibria. Since even the existence of
nontrivial stationary points is often difficult to accomplish, extracting their
index information is usually out of reach. In this paper we establish a
computer-assisted proof approach to determining relative indices of stationary
states. We introduce the general framework and then focus on three example
problems described by partial differential equations to show how these ideas
work in practice. Based on a rigorous implementation, with accompanying code
made available, we determine the relative indices of many stationary points.
Moreover, we show how forcing results can be then used to prove theorems about
connecting orbits and traveling waves in partial differential equations.Comment: 30 pages, 4 figures. Revised accepted versio
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Optimal Constants in the Theory of Sobolev Spaces and PDEs
Recent research activities on sharp constants and optimal inequalities have shown their impact on a deeper understanding of geometric, analytical and other phenomena in the context of partial differential equations and mathematical physics. These intrinsic questions have applications not only to a-priori estimates or spectral theory but also to numerics, economics, optimization, etc
Analytical solutions to nonlinear differential equations arising in physical problems
Nonlinear partial differential equations are difficult to solve, with many of the approximate solutions in the literature being numerical in nature. In this work, we apply the Homotopy Analysis Method to give approximate analytical solutions to nonlinear ordinary and partial differential equations. The main goal is to apply different linear operators, which can be chosen, to solve nonlinear problems. In the first three chapters, we study ordinary differential equations (ODEs) with one or two linear operators. As we progress, we apply the method to partial differential equations (PDEs) and use several linear operators. The results are all purely analytical, meaning these are approximate solutions that we can evaluate at points and take their derivatives. Another main focus is error analysis, where we test how good our approximations are. The method will always produce approximations, but we use residual errors on the domain of the problem to find a measure of error. In the last two chapters, we apply similarity transforms to PDEs to transform them into ODEs. We then use the Homotopy Analysis Method on one, but are able to find exact solutions to both equations
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