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In the present paper, we study a class of nonlinear time‐fractional pseudo‐parabolic equations with bounded time‐delay. Firstly, we analyze the well‐posedness of the problem by the well‐known Galerkin's method and some compactness criteria. Next, the existence of the stationary solutions is established by utilizing a lemma of ours, which is a corollary of the Brouwer fixed point theorem. Finally, with regard to the stability analysis of the stationary solutions, we prove the local stability via the direct method of energy estimations and show the asymptotic stability by the construction of appropriate Lyapunov functionals. [ABSTRACT FROM AUTHOR]

In this work, we construct a new family of inverse iterative numerical technique for extracting all roots of nonlinear equation simultaneously. Convergence analysis verifies that the proposed family of methods has local 10th-order convergence. Among the test models investigated are blood rheology, a fractional nonlinear equation model, fluid permeability in biogels, and beam localization models. In comparison to other methods, the family of inverse simultaneous iterative techniques gets initial estimations to exact roots within a given tolerance while using less function evaluations in each iterative step. Numerical results, basins of attraction for fractional nonlinear equation, residual graphs are presented in detail for the simultaneous iterative techniques. The newly developed simultaneous iterative techniques were thoroughly investigated and proven to be efficient, robust, and authentic in their domain. [ABSTRACT FROM AUTHOR]

The main goal of this manuscript is to study the existence results in the Bessel Potential space for the following convection‐reaction fractional problem involving by distributional Riesz fractional derivative, −Dα.(Dαu(x))−divαbφ(u(x))=f(x,u(x),Dαu(x))inΩ,u=0onℝd/Ω,$$ \left\{\begin{array}{cc}-{D}^{\alpha }.\left({D}^{\alpha }u(x)\right)- di{v}^{\alpha}\left(b\varphi \left(u(x)\right)\right)=f\left(x,u(x),{D}^{\alpha }u(x)\right)\kern0.60em & \kern0.1em \mathrm{in}\kern0.51em \Omega, \\ {}u=0\kern0.30em & \kern0.1em \mathrm{on}\kern0.51em {\mathbb{R}}^d/\Omega, \end{array}\right. $$where Ω⊂ℝd$$ \Omega \subset {\mathbb{R}}^d $$ is a bounded open set with a Lipschitz boundary, α∈(0,1)$$ \alpha \in \left(0,1\right) $$ and d>2α$$ d>2\alpha $$. To reach our goal, we use the application of the Schauder fixed point theory with some assumption on the nonlinear terms φ$$ \varphi $$ and f$$ f $$. [ABSTRACT FROM AUTHOR]

The space–time fractional nonlinear Tzitzeica–Dodd–Bullough (TDB) equation is an important model in nonlinear optics, quantum field theory, plasma physics, solid-state physics among other domains. Through the Painlevé and fractional wave transformations, the space–time fractional nonlinear TDB equation has been converted to a nonlinear equation. Broad-spectral and standard closed-form soliton solutions in the form of exponential, rational, trigonometric, and hyperbolic functions with free parameters have been achieved using the improved Bernoulli sub-equation function (IBSEF) approach. The solutions established in this article are comprehensive and sophisticated, and the results found in the literature are a special case of the results obtained. The effect of the fractional parameter μ on the wave profiles of the phenomena has been examined by defining 3D, 2D, and contour plots of the soliton solutions, and it is found that the fractional parameter has a significant impact. The results attained demonstrate that the IBSEF approach is compatible, useful, and capable of providing wide-spectral analytical wave solutions to fractional nonlinear models arising in optics, mathematical physics, and engineering. [ABSTRACT FROM AUTHOR]