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Bulletin of the American Mathematical Society

Solution of this over-determined system of algebraic equations with the help of computer algebra yields:. Using 6 into the solution 9 and substituting Cases 1—9, we obtain abundant traveling wave solutions including soliton, singular soliton, periodic solution, etc. If we set specific values of A and B , various known solutions can be rediscovered.

For example, soliton, periodic and complex solutions can be derived from the traveling wave solutions 10 :. These are exact bell-type soliton solutions.

MATH 4500 - Methods of Partial Differential Equations of Mathematical Physics

Solution 12 is the singular soliton solution. Singular solitons are another kind of solitary waves that appear with a singularity, usually infinite discontinuity Wazwaz Singular solitons can be connected to solitary waves when the center position of the solitary wave is imaginary Drazin and Johnson Therefore it is not irrelevant to address the issue of singular solitons.

This solution has spike and therefore it can probably provide an explanation to the formation of Rogue waves. When A and B receive particular values, different known solutions will be rediscovered. For example:. Periodic traveling waves play an important role in numerous physical phenomena, including reaction—diffusion—advection systems, impulsive systems, self-reinforcing systems, etc. Mathematical modelling of many intricate physical events, for instance biology, chemistry, physics, mathematical physics and many more phenomena resemble periodic traveling wave solutions.

Setting particular values of the free parameters involved in solutions 10 — 38 abundant soliton, singular solitons, periodic solutions and general solitary wave solutions can be found. It is noteworthy to refer that some of our obtained solutions are identical to the solutions achieved by Zayed and Al-Joudi which validate our solutions and some are new.

These solutions might be much important for the explanation of some special physical phenomena. The solitary wave ansatz method can be applied to high-dimensional or coupled nonlinear PDEs in mathematical physics. We expect the attained solutions may be useful for further numerical analysis and may help the researchers to explain complex physical phenomena. The solutions obtained in this article are in more general forms and many known solutions to this equation are only special cases.

Mathematical Physics Equations - EqWorld

This study shows that the proposed ansatz is reliable, effective and computerized which permit us to carry out complicated and tiresome algebraic calculation and giving new solutions to the applied equation. This ansatz can be applied to both single equation and coupled equations to establish further new solutions for other kinds of nonlinear partial differential equations. This work was carried out in collaboration between the authors. Both authors have a good contribution to design the study, and to perform the analysis of this research work.


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Both authors read and approved the final manuscript. The authors also would like to express their gratitude to the anonymous referees for their valuable comments and suggestions.

Mathematical Physics

National Center for Biotechnology Information , U. Published online Jan 7. Ali Akbar and Norhashidah Hj. Ali Akbar. Norhashidah Hj.

Author information Article notes Copyright and License information Disclaimer. Ali Akbar, Email: moc. Corresponding author. Received Nov 9; Accepted Dec Abstract In this article, we introduce an ansatz involving exact traveling wave solutions to nonlinear partial differential equations. From there, more advanced concepts are developed in detail and with great precision; moreover, theorems are often approached through the study of special simpler cases, before being proved in their full generality, and are applied to many particular physical problems.

After deriving the fundamental equations, the author provides illuminating expositions of such topics as Riemann's method, Lebesgue integration of multiple integrals, the equation of heat conduction, Laplace's equation and Poisson's equation, the theory of integral equations, Green's function, Fourier's method, harmonic polynomials and spherical functions, and much more. For this third edition, various improvements in style and clarifications of the presentations were made, including a simplification of the theory of multiple Lebesgue integrals and greater precision in the proof of the Fourier method.

Finally, the translation is both idiomatic as well as accurate, making the vast amount of information in this book more readily accessible to the English reader. Book Reg. Product Description Product Details The classical partial differential equations of mathematical physics, formulated by the great mathematicians of the 19th century, remain today the basis of investigation into waves, heat conduction, hydrodynamics, and other physical problems. Reprint of the Pergamon Press, edition.


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