Volume 22, Issue 5

In this work, we study a numerical approach for studying a nonlinear model of fractional optimal control problems (FOCPs). We have taken the fractional derivative in a dynamical system of FOCPs, which is in Liouville–Caputo sense. The presented scheme is a grouping of an operational matrix of integrations for Jacobi polynomials and the Ritz method. The proposed approach converts the FOCP into a system of nonlinear algebraic equations, which significantly simplify the problem. Convergence analysis of the scheme is also provided. The presented method is verified on the two illustrative examples to show its accuracy and applicability. Distinct special cases of Jacobi polynomials are considered as a basis to solve the FOCPs for comparison purpose. Further, tables and figures are employed to demonstrate the derived numerical results. The numerical results by the present method are also compared with some other techniques.

Based on full domain partition, three parallel iterative finite-element algorithms are proposed and analyzed for the Navier–Stokes equations with nonlinear slip boundary conditions. Since the nonlinear slip boundary conditions include the subdifferential property, the variational formulation of these equations is variational inequalities of the second kind. In these parallel algorithms, each subproblem is defined on a global composite mesh that is fine with size h on its subdomain and coarse with size H ( H  ≫  h ) far away from the subdomain, and then we can solve it in parallel with other subproblems by using an existing sequential solver without extensive recoding. All of the subproblems are nonlinear and are independently solved by three kinds of iterative methods. Compared with the corresponding serial iterative finite-element algorithms, the parallel algorithms proposed in this paper can yield an approximate solution with a comparable accuracy and a substantial decrease in computational time. Contributions of this paper are as follows: (1) new parallel algorithms based on full domain partition are proposed for the Navier–Stokes equations with nonlinear slip boundary conditions; (2) nonlinear iterative methods are studied in the parallel algorithms; (3) new theoretical results about the stability, convergence and error estimates of the developed algorithms are obtained; (4) some numerical results are given to illustrate the promise of the developed algorithms.

In this paper, we study the existence of ground state solutions for the following fractional Kirchhoff–Schrödinger–Poisson systems with general nonlinearities: { ( a + b [ u ] s 2 )   ( − Δ ) s u + u + ϕ ( x ) u = ( | x | − μ ∗ F ( u ) ) f ( u ) in   ℝ 3   , ( − Δ ) t ϕ ( x ) = u 2 in   ℝ 3   , $$\left\{\begin{array}{ll}\left(a+b{\left[u\right]}_{s}^{2}\right)\,{\left(-{\Delta}\right)}^{s}u+u+\phi \left(x\right)u=\left({\vert x\vert }^{-\mu }\ast F\left(u\right)\right)f\left(u\right)\hfill & \mathrm{in}\text{\ }{\mathrm{&#x211d;}}^{3}\,\text{,}\hfill \\ {\left(-{\Delta}\right)}^{t}\phi \left(x\right)={u}^{2}\hfill & \mathrm{in}\text{\ }{\mathrm{&#x211d;}}^{3}\,\text{,}\hfill \end{array}\right.$$ where [ u ] s 2 = ∫ ℝ 3 | ( − Δ ) s 2 u | 2   d x = ∬ ℝ 3 × ℝ 3 | u ( x ) − u ( y ) | 2 | x − y | 3 + 2 s   d x d y   , $${\left[u\right]}_{s}^{2}={\int }_{{\mathrm{&#x211d;}}^{3}}{\vert {\left(-{\Delta}\right)}^{\frac{s}{2}}u\vert }^{2}\,\mathrm{d}x={\iint }_{{\mathrm{&#x211d;}}^{3}{\times}{\mathrm{&#x211d;}}^{3}}\frac{{\vert u\left(x\right)-u\left(y\right)\vert }^{2}}{{\vert x-y\vert }^{3+2s}}\,\mathrm{d}x\mathrm{d}y\,\text{,}$$ s , t ∈ ( 0 , 1 ) $s,t\in \left(0,1\right)$ with 2 t + 4 s > 3 , 0 < μ < 3 − 2 t , $2t+4s{ >}3,0{< }\mu {< }3-2t,$ f : ℝ 3 × ℝ → ℝ $f:{\mathrm{&#x211d;}}^{3}{\times}\mathrm{&#x211d;}\to \mathrm{&#x211d;}$ satisfies a Carathéodory condition and (−Δ) s is the fractional Laplace operator. There are two novelties of the present paper. First, the nonlocal term in the equation sets an obstacle that the bounded Cerami sequences could not converge. Second, the nonlinear term f does not satisfy the Ambrosetti–Rabinowitz growth condition and monotony assumption. Thus, the Nehari manifold method does not work anymore in our setting. In order to overcome these difficulties, we use the Pohozǎev type manifold to obtain the existence of ground state solution of Pohozǎev type for the above system.

In this paper, we examine a coupled system of fractional integrodifferential equations of Liouville-Caputo form with nonlinearities depending on the unknown functions, as well as their lower-order fractional derivatives and integrals supplemented with coupled nonlocal and Erdélyi-Kober fractional integral boundary conditions. We explain that the topic discussed in this context is new and gives more analysis into the research of coupled boundary value problems. We have two results: the first is the existence result of the given problem by using the Leray-Schauder alternative, whereas the second referring to the uniqueness result is derived by Banach’s fixed-point theorem. Sufficient examples were also supplemented to substantiate the proof, and some variations of the problem were discussed.

In this paper, we consider the optimization problems of exact controllability for a new class of fractional impulsive partial stochastic differential systems with state-dependent delay in Hilbert spaces. By utilizing suitable fixed point approach without imposing severe compactness condition on the operators, the theory of analytic sectorial operators, stochastic analysis, and the Hausdorff measure of noncompactness, some sufficient conditions are derived for achieving the required results. Finally, an example is provided to illustrate the obtained theory.

A numerical algorithm for the solution of an inverse coefficient problem for nonstationary, nonlinear production-destruction type model is proposed and tested on an example of the Lorenz’63 system. With an ensemble of adjoint problem solutions, the inverse problem is transformed into a quasi-linear matrix problem and solved with Newton-type algorithm. Two different ways of the adjoint ensemble construction are compared. In the first one, a trigonometric basis is used. In the second one in situ measurements are taken into account. Local convergence properties of the algorithm are studied numerically to find out when the use of more data can lead to the degradation of the reconstruction results.

In this paper, we establish the existence of mild solutions for nonlocal fractional semilinear differential inclusions with noninstantaneous impulses of order α ∈ (1,2) and generated by a cosine family of bounded linear operators. Moreover, we show the compactness of the solution set. We consider both the case when the values of the multivalued function are convex and nonconvex. Examples are given to illustrate the theory.