International Journal of Differential Equations

The fractional differential equations (FDEs) are ubiquitous in mathematically oriented scientific fields, such as physics and engineering. Therefore, FDEs have been the focus of many studies due to their frequent appearance in several applications such as physics, engineering, signal processing, systems identification, sound, heat, diffusion, electrostatics and fluid mechanics, and other sciences. The perusal of these nonlinear physical models through wave solutions analysis, corresponding to their FDEs, has a dynamic role in applied sciences. In this paper, the exp-function method and the rational -expansion method are presented to establish the exact wave solutions of the space-time fractional Drinfeld–Sokolov–Wilson system in the sense of the conformable fractional derivative. The fractional Drinfeld–Sokolov–Wilson system contains fractional derivatives of the unknown function in terms of all independent variables. This system describes the shallow water wave models in fluid mechanics. These presented methods are a powerful mathematical tool for solving nonlinear conformable fractional evolution equations in various fields of applied sciences, especially in physics.

Using the Schauder fixed point theorem, we prove the existence of impulsive fractional differential equations using Hilfer fractional derivative and nearly sectorial operators in this paper. We’ve gone over the two scenarios where the related semigroup is compact and noncompact for this purpose. We also go over an example to back up the main points.

The work addresses the exponential moment stability of solutions of large systems of linear differential Itô equations with variable delays by means of a modified regularization method, which can be viewed as an alternative to the technique based on Lyapunov or Lyapunov-like functionals. The regularization method utilizes the parallelism between Lyapunov stability and input-to-state stability, which is well established in the deterministic case, but less known for stochastic differential equations. In its practical implementation, the method is based on seeking an auxiliary equation, which is used to regularize the equation to be studied. In the final step, estimation of the norm of an integral operator or verification of the property of positivity of solutions is performed. In the latter case, one applies the theory of positive invertible matrices. This report contains a systematic presentation of how the regularization method can be applied to stability analysis of linear stochastic delay equations with random coefficients and random initial conditions. Several stability results in terms of positive invertibility of certain matrices constructed for general stochastic systems with delay are obtained. A number of verifiable sufficient conditions for the exponential moment stability of solutions in terms of the coefficients for specific classes of Itô equations are offered as well.

The findings indicate an application of a new method of expansion of the forms and to determine the solutions for wave of the solitary nature in the -dimensional modified form for nonlinear integro-partial differential equations. By using this strategy, we acquired solutions of wave which has a solitary nature that have been solved for three different kinds: hyperbolic, trigonometric, and rational functions. As a result, we obtained different forms of solutions which are new, effective, and powerful to illustrate the solitary nature of waves. The physical and geometrical interpretations have been shown using software in 2 and 3-dimensional surfaces. The obtained results have applications in mathematical and applied sciences. It can also solve different nonlinear integro-partial differential equations which have different applications in physical phenomena using this new method. It has many applications to solve the nonlinear nature of the physical world.

In this paper, we define the space of functions -bounded variation on the plane and endow it with a norm under which it is a Banach space. In addition, we study some nonlinear integral equations and providing conditions for the functions and kernel involved in such equations under which we guarantee the existence and uniqueness in the space of functions of bounded variation in the sense of Shiba on the plane, .

This paper discusses some sufficient conditions for oscillatory behavior of even-order half-linear neutral differential equation. An example is given to illustrate the main result.