Computational and Applied Mathematics最新文献

This paper introduces some Krylov subspace methods utilizing the s-step technique. The variable s-step technique is applied to CGS and BiCG algorithms, and extended to the BiCGstab algorithm as an intermediate state between this two algorithms. By proposing the use of the s parameter as a variable, these algorithms become adaptable. To enhance stability, a regularization technique is incorporated. Through the integration of these techniques, stable algorithms are developed. Numerical examples are provided to demonstrate the efficacy and quality of the proposed algorithms.

In this article, we present a fully discrete Crank–Nicolson Galerkin finite element method for solving the two-dimensional nonlinear extended-Fisher–Kolmogorov equation: (u_t + gamma Delta ^2 u -Delta u -u +u^{3} = 0.) The boundedness of the numerical solution in the maximum norm, unique solvability, and related convergence results in (L^2) and (L^{infty })-norms are studied in detail. Also, a new linearized Crank–Nicolson Galerkin modification scheme is designed and error estimate without any time step restrictions is established. Finally, some computational experiments in one and two dimension cases are provided to illustrate the efficacy of our method and to confirm the theoretical results.

A novel framework is proposed in this research based on multilevel method to solve the optimal control problem. In recent dacades, the mathematical theory of optimal control has rapidly developed into an important and separate field of applied mathematics. The solution of nonlinear partial differential equations is considerably difficult, and the theory of their optimal control is still an open field in many respects. These optimization problems have found diverse applications in various sciences including electrical engineering, mechanical engineering, and aerospace. Current methods for solving this class of optimal control problems usually fall into two classes: discrete-then-optimization or optimization-then-discrete approaches. The proposed approach, however, does not require discretization as it involves rewriting the optimal control problem as a multi-objective optimization problem followed by its solution with a feedforward single-layer artificial neural network based on learning through by the multi-level Levenberg–Marquardt method. Moreover, the convergence of the approach was discussed and some numerical results are presented.

Let G be a connected graph. The distance between two vertices u and v in G, denoted by (d_G(u,v)), is the number of edges in a shortest path from u to v, while the distance between an edge (e = xy) and a vertex v in G is (d_G(e,v) = min {d_G(x,v),d_G(y,v)}). For an edge (e in E(G)) and a subset S of V(G), the representation of e with respect to (S={x_1,ldots ,x_k}) is the vector (r_G(e|S) =(d_1,ldots ,d_k)), where (d_i=d_G(e,x_i)) for (i in [k]). If (r_G(e|S)ne r_G(f|S)) for every two adjacent edges e and f of G, then S is called a local edge metric generator for G. The local edge metric dimension of G, denoted by (mathrm{edim_ell }(G)), is the minimum cardinality among all local edge metric generators in G. For two non-trivial graphs G and H, we determine (mathrm{edim_ell }(G diamond H)) in the edge corona product (G diamond H) and we determine (mathrm{edim_ell }(Gcirc H)) in the corona product (Gdiamond H). We also formulate the problem of computing (mathrm{edim_ell }(G)) as an integer linear programming model.

At present, a large number of fractional differential models of migration processes in soils are developed. Their practical application largely depends on the possibility to determine the values of their parameters. In this regard, we study the possibility of recovering the values of parameters for one such generalized model from noised data in order to assess the threshold of measurement accuracy, beyond which the complication of a model leads to an inability to distinguish its solutions from the solutions of simpler models. We consider the 1D fractional-order model of water head dissipation in water-saturated soil with linear deformation that includes the Caputo–Fabrizio derivative with respect to the time variable and the Riemann–Liouville derivative with respect to the space variable. Direct problems for this model are proposed to be solved by an optimized computational procedure based on a finite-difference scheme. Inverse problems of model’s parameter identification are solved using a multi-threaded Particle Swarm Optimization technique. The results of computational experiments showed that the values of model parameters can be restored with less than (10%) relative error for the number of input water head values equal to 1000 and the level of noise less than (5%). Our results also show that the order of the Riemann–Liouville derivative can be with an average relative error of less than (3%) restored even at (10%) level of noise and 40 input values, when the accuracy of other parameters’ restoration drops significantly.

Recently, finding extremal structures of graphs on Sombor index has received a lot of attention. The Sombor (SO) index of a graph G is defined by the sum of weights (sqrt{deg_{G}(u)^{2}+deg_{G}(v)^{2}}) over all edges uv of G, where (deg_{G}(u)) stands for the degree of vertex u in G. In this article, we obtain a lower bound on Sombor index of trees with a given order and total domination number, and characterize the trees achieving the bound.

We derive the prolongation formula of the one-parameter Lie point transformations to the Hilfer fractional derivative and show that the existing prolongation formula for the Riemann Liouville and Caputo fractional derivatives are special cases of the proposed formula, corresponding to the type parameter (gamma =0) and (gamma =1), respectively. The applicability of the proposed formula is demonstrated by deriving the Lie point symmetries of the time-fractional heat equation, the fractional Burgers equation, and the fractional KdV equation in Hilfer’s sense. We use the obtained Lie point symmetries to find the similarity variables and transformations. Using the similarity transformations, we show that each is converted into a nonlinear fractional ordinary differential equation with a new independent variable. The fractional derivative in the reduced equation can be either the Hilfer-type modification of the Erdélyi Kober fractional derivative or the Hilfer fractional derivative itself. We demonstrate that the exact solution of the time-fractional differential equation in the Hilfer sense can be reduced to the exact solutions of the corresponding time-fractional differential equations in the Riemann–Liouville and Caputo senses by setting the type parameter to (gamma =0) and (gamma =1), respectively.