In the paper, we investigate meromorphic solutions in ℂ^m of the following partial differential equations f^n∂ _z_1(f)+P_d(f)=Pe^α, d≤ n-1 and f^n∂ _z_1(f)+P_d(f)=P_1e^α _1+P_2e^α _2, d≤ n-2, where P_d(f) is a differential polynomial in f of degree d with rational function as its coefficients of degree, n≥ 2 is an integer, P, P_1 and P_2 are non-zero rational functions, α , α _1 and α _2 are non-constant polynomial in ℂ^m such that ∂ _z_1(α )≢0 and ∂ _z_1(α _i)≢0 , i=1,2 . Our findings extend and improve the results of Liao and Ye (2014) and Zhang et al. (2017) into ℂ^m with (ℂ^m)≥ 2 .
In this paper, we design and analyze a modified spectral collocation method based on Lagrange interpolation polynomials for solving a class of nonlinear fractional weakly singular Volterra integro–differential equations. The fractional derivative is considered in the Caputo sense. The proposed scheme employs sixth–kind Chebyshev polynomials, a family of symmetric orthogonal polynomials, to construct the novel basis functions. We reformulate the original equation and its initial condition into an equivalent integro–differential system by exploiting the relationship between the Riemann–Liouville integral operator and the Caputo derivative. Newton–Raphson iteration method is then applied to compute the numerical solutions and approximate fractional derivatives simultaneously. The convergence of the proposed method is analyzed in both the L^∞ – and weighted L^2 –norms. Our results demonstrate that the errors in the numerical solutions and approximate fractional derivatives decay exponentially. Numerical experiments validate the effectiveness and accuracy of the proposed scheme.
We consider a new class of operators, termed polynomially quasi-(m, C)-expansive operators, and examine its fundamental properties. The resulting family of operators unifies and extends several important notions in operator theory, including m-expansive and n-quasi-m-expansive operators, by incorporating polynomial constraints into the defining relations. This framework provides additional flexibility for analyzing operator behavior and reveals new structural properties. We develop the foundational theory of this class by establishing its basic properties, investigating its stability under natural operations, and examining its invariance when restricted to closed invariant subspaces. Furthermore, we study the relationships between polynomially quasi- (m,𝒞) -expansive operators and existing operator classes, providing illustrative examples to demonstrate the properness of the inclusions obtained. We also derive spectral properties of these operators, highlighting the influence of polynomial constraints on their spectral structure. The proposed class offers a new perspective on expansive-type operators, with its polynomial structure providing a basis for further investigation.
Surface sediments were collected from eight representative small- and medium-sized rivers in Binzhou City, located in the hinterland of the Yellow River Delta, to determine the concentrations of Cu, Zn, Ni, Pb, Cr, As, and Hg. Zn and Cr were the dominant metals, whereas Hg showed the lowest concentration, following the order Zn > Cr > Pb > Ni > Cu > As > Hg. Except for Pb, which marginally exceeded its adopted regional soil-derived reference value, the mean concentrations of the other elements were below or close to the corresponding soil reference values, indicating generally low heavy metal enrichment. Spatial differences between the Haihe and Huaihe River basins were mainly observed for Cu, Zn, Cr, and As, whereas Pb showed no significant difference after correction. Cu, Zn, Ni, Pb, Cr, and As showed similar covariation patterns, whereas Hg exhibited a relatively distinct distribution pattern. Local exceedances of TECs for Cu, Ni, Pb, Cr, and As, together with local PEC exceedances for Cr, supported continued site-specific monitoring. Geo-accumulation index (Igeo) values were predominantly below 0, and the potential ecological risk index (RI) ranged from 30.56 to 102.27, indicating low overall ecological risk; Hg was the major contributor to integrated risk. Overall, heavy metal pollution in river sediments of Binzhou City was not pronounced, but Cu, Ni, Cr, As, and Hg in certain reaches warrant continued attention.
In this paper, we study the integrated radial basis function based on the differential quadrature (IRBF-DQ) method to solve the nonlinear time-fractional partial differential equations with multi-fractional orders. First, the multi-term Caputo time fractional derivative is discretized for fractional order in (1, 2) . Then IRBF-DQ is used for discretization of the spatial variables. This method is highly demanding because of its mesh-less nature and ease of implementation in high dimensions and complex geometries. We utilize the evaluation of the previous time level for the nonlinear terms that satisfy Lipschitz condition. The time fractional derivatives which have been described in Caputo-s sense are approximated by schemes of orders O(τ ^3-β _k) for β _k∈ (1,2) . We also prove that the time discrete scheme is unconditionally stable and the convergence order of the time discrete scheme is O(τ ^3-min _0≤ k≤ mβ _k) . Numerical problems are yielded to judge the behavior of the presented method for fractional order. The numerical experiments will be verified to demonstrate the effectiveness of our method for solving one- and two-dimensional regions of multi-term time fractional partial differential equations. Numerical simulation is carried out for regular (uniform and nonuniform points) and irregular domains.
