a b s t r a c tIn this paper, we obtain an Ostrowski and Ostrowski-Grüss type inequality on time scales, which not only provides a generalization of the known results on time scales, but also give some other interesting inequalities as special cases.

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The nonlinear dynamics and propagation of ion acoustic waves in a relativistic and ideal plasmas, which have the pressure variation of electrons and ions and degenerate electrons, are investigated using the analytic solution of KdV type equations performed applying (G′/G)-expansion and (G′/G,1/G)-expansion methods. The effects of various parameters, such as phase velocity of the ion acoustic wave, the ratio of ion temperature to electron temperature, normalized speed of light, electron and ion streaming velocities, arbitrary and integration constants, on the soliton dynamics are studied. We have found that dim and hump solitons and their amplitudes, widths and dynamics strongly depend on these plasma parameters and integration constants. The source term μ plays also a vital role in the formation of the solitons. Moreover, it is also found that the observed solitary wave solution can be excited from hump soliton to dip soliton. This dramatic change of the solitons can occur due to the various values of the integration constants and ion streaming velocities. Finally, it is important to note that the analytic solutions of the nonlinear equation, reported in this study, could be used to explain the structures of solitons in the astrophysical space and in laboratory plasmas.

Nonlinear Drinfeld-Sokolov system is studied analytically by using four different methods ((G′/G)-expansion, direct algebraic, different form of the (G′/G)-expansion methods, and direct integration) and the results are found numerically. New exact and numeric solutions are given and the comparison of the results obtained from these different methods, methods themselves and numerical results are discussed in detail. It is found that the (G′/G)-expansion and different form of the (G′/G)-expansion methods are really coincide and effective methods in the view of finding different solutions that cannot be obtained by using the direct integration for Drinfel-Sokolov system.

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We obtain an exact solution for the Einstein's equations with cosmological constant coupled to a scalar, static particle in static, "spherically" symmetric background in (2 + 1)-dimensions.

Abstract:The wave properties in a dusty space plasma consisting of positively and negatively charged dust as well as distributed nonisothermal electrons are investigated by using the exact traveling wave solutions of the Schamel-KdV equation. The analytic solutions are obtained by the different types (G ′ /G)-expansion methods and direct integration. The nonlinear dynamics of ion-acoustic waves for the various values of phase speed V p , plasma parameters α, σ , and σ d , and the source term µ are studied. We have observed different types of waves from the different analytic solutions obtained from the different methods. Consequently, we have found the discontinuity, shock or solitary waves. It is also concluded that these parameters play an important role in the presence of solitary waves inside the plasma. Depending on plasma parameters, the discontinuity wave turns into solitary wave solution for the certain values of the phase speed and plasma parameters. Additionally, exact solutions of the Schamel-KdV equation may also be used to understand the wave types and properties in the different plasma systems. Farklı Metodlar ile Schamel-KdV denkleminin Analitik Çözümleri: Tozlu Uzay Plazmasına UygulanmasıAnahtar Kelimeler Schamel-KdV denklemi, Tozlu uzay plazması, Sok dalgası, SolitonÖzet:İçerisinde negatif ve pozitif yüklü tozların yanında dagılmış izotermal olmayan elektronlar barındıran tozlu uzay plazmasındaki dalganın özellikleri, Schamel-KdV denklemlerinin tam ilerleyen dalga çözümleri kullanılarak incelenmiştir. Analitik çözümler, (G ′ /G)-genişleme methodunun farklı tipleri ve direk integrasyon kullanılarak bulunmuş-tur.İon-akustik dalgasının lineer olmayan dinamigi, faz hızının V p , plazma parametreleri α, σ , ve σ d , ve kaynak terimi µ'nun farklı degerleri için çalışılmıştır. Bunun sonucunda, farklı methodlardan elde edilen farklı analitik çözümler ile farklı türden dalgalar gözlemledik ve süreksiz,şok veya soliton dalgası bulduk. Aynı zamanda, yukarıda verilen parametrelerin plazma içerisinde soliton tipi dalgaların oluşmasında önemli bir rol oynadıgı sonucuna ulaşılmıştır. Bu parametrelere baglı olarak süreksiz dalga plazma parametrelerinin ve faz hızının belli degerleri için soliton tipi bir dalgaya donüşür. Bunlara ek olarak, Schamel-KdV denkleminin tam analitik çözümleri, verilen bir plazmanın özelliklerinin ve dalga tiplerinin anlaşılması için farklı plazma sistemlerine de uygulanabilir.