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This paper is concerned with an efficient numerical method for a class of parabolic integro-differential equations with weakly singular kernels. Due to the presence of the weakly singular kernel, the exact solution has singularity near the initial time t = 0 . A generalized Crank-Nicolson-type scheme for the time discretization is proposed by designing a product integration rule for the integral term, and a compact difference approximation is used for the space discretization. The proposed method is constructed on the uniform time mesh, but it can still achieve the second-order convergence in time for weakly singular solutions. The unconditional stability and convergence of the method is proved and the optimal error estimate in the discrete L 2 -norm is obtained. The error estimate shows that the method has the second-order convergence in time and the fourth-order convergence in space. The extension of the method to two-dimensional problems is also discussed. A simple comparison is made with several existing methods. Numerical results confirm the theoretical analysis result and show the effectiveness of the proposed method.

Purpose This paper aims to discuss the stagnation-point flow and heat transfer for power-law fluid pass through a stretching surface with heat generation effect. Unlike the previous considerations about the research on stagnation-point flow, the process of heat transfer and the convective heat transfer boundary condition use the modified Fourier’s law in which the heat flux is power-law-dependent on velocity gradient. Design/methodology/approach The similarly transformation is used to convert the governing partial differential equations into a series of ordinary differential equations which are solved analytically by using the differential transform method and the base function method. Findings The variations of the velocity and temperature fields for different specific related parameters are graphically discussed and analyzed. There is a special phenomenon that all the velocity profiles converge from the initial value of velocity to stagnation parameter values. And the larger power-law index enhancesthe momentum diffusion. A significant phenomenon can be observed that the larger power-law index causes a decline in the heat flux. This influence indicates that the higher viscosity restricts the heat transfer. Furthermore, both velocity gradient and temperature gradient play an indispensable role in the processes of heat transfer. Originality/value This paper researches the process of heat transfer of stagnation-point flow ofpower-law magneto-hydro-dynamical fluid over a stretching surface with modified convective heat transfer boundary condition.

β-Catenin is an important co-factor which binds multiple transcriptional molecules and mediates fibrogenic signaling pathways. Its role in kidney transplantation is unknown. We quantified binding of β-catenin within renal tubular epithelial cells to transcription factors, TCF1 and FoxO1, using a proximity ligation assay in 240 transplanted kidneys, and evaluated their pathological and clinical outcomes. β-Catenin/FoxO1 binding in 1-month protocol biopsies inversely correlated with contemporaneous chronic fibrosis, subsequent inflammation. and inflammatory fibrosis (P < .001). The relative binding of β-catenin/TCF1 versus β-catenin/FoxO1 (TF ratio) was the optimal biomarker, and abnormal in diverse fibrotic transplant diseases. A high 1-month TF ratio was followed by greater tubular atrophy and interstitial fibrosis scores, cortical inflammation, renal impairment, and proteinuria at 1 year (n = 131, all P < .001). The TF ratio was associated with reduced eGFR (AUC 0.817), mild fibrosis (AUC 0.717), and moderate fibrosis (AUC 0.769) using receiver operating characteristic analysis. An independent validation cohort (n = 76) confirmed 1-month TF was associated with 12-month moderate fibrosis (15.8% vs. 2.6%, P = .047), however, not with other outcomes or 10-year graft survival, which limits generalizabilty of these findings. In summary, differential binding of β-catenin to TCF1 rather than FoxO1 in renal tubular cells was associated with the fibrogenic response in transplanted kidneys.

A high-order compact finite difference method on nonuniform time meshes is proposed for solving a class of variable coefficient reaction–subdiffusion problems. The solution of such a problem in general has a typical weak singularity at the initial time. Alikhanov’s high-order approximation on a uniform time mesh for the Caputo time fractional derivative is generalised to a class of nonuniform time meshes, and a fourth-order compact finite difference scheme is used for approximating the spatial variable coefficient differential operator. A full theoretical analysis of the stability and convergence of the method is given for the general case of the variable coefficients by developing an analysis technique different from the one for the constant coefficient problem. Taking the weak initial singularity of the solution into account, a sharp error estimate in the discrete $$L^{2}$$ -norm is obtained. It is shown that the proposed method attains the temporal optimal second-order convergence provided a proper mesh parameter is employed. Numerical results demonstrate the sharpness of the theoretical error analysis result.

This paper is concerned with the construction and analysis of a high-order compact finite difference method for a class of time-fractional sub-diffusion equations under the Robin boundary condition. The diffusion coefficient of the equation may be spatially variable and the time-fractional derivative is in the Caputo sense with the order α ∈ ( 0 , 1 ) . A ( 3 − α ) th-order numerical formula (called the L2 formula here) without any sub-stepping scheme for the approximation at the first-time level is applied to the discretization of the Caputo time-fractional derivative. A new fourth-order compact finite difference operator is constructed to approximate the variable coefficient spatial differential operator under the Robin boundary condition. By developing a technique of discrete energy analysis, the unconditional stability of the proposed method and its convergence of ( 3 − α ) th-order in time and fourth-order in space are rigorously proved for the general case of variable coefficient and for all α ∈ ( 0 , 1 ) . Further approximations are considered for enlarging the applicability of the method while preserving its high-order accuracy. Numerical results are provided to demonstrate the theoretical analysis results.