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This study provides a quantitative analysis of the effects of neglecting the quadratic-gradient term in solving the diffusion equation governing the transient pressure distribution during high pressuregradient injection of compressible liquids in porous media. Mathematical solutions of the twodimensional cylindrical-symmetry nonlinear diffusion equation are derived by using the Laplace transform. A fully penetrating well bore in a homogeneous and isotropic porous medium is considered. The analysis accounts for well bore storage and incorporates a wide range of boundary conditions. Analytical early-and late-time solutions are also presented for some cases. Quantitative deviations from existing linear solutions are related to a dimensionless group, a, which is proportional to the fluid compressibility; the higher the magnitude of a, greater is the deviation of the nonlinear solutions from the linear ones. The linear pressure and rate solutions are generally within 0.5% of the corresponding nonlinear solutions for the constant pressure inner boundary. However, for the constant dischargerate condition, the error may be as high as 10% (within the ranges of a and dimensionless radius and time considered). The error may be even higher for higher injection rates in flow systems with smaller transmissivity. Stehfest, H., Numerical inversion ofLaplace transforms, Commun. ACM, 13(1), 47-49, 1970. Wang, Y., and M. B. Dusseault, The effect of quadratic gradient terms on the borehole solution in poroelastic media, Water

In this article, sufficient conditions for the existence of extremal solutions to nonlinear boundary value problem (BVP) of fractional order differential equations (FDEs) are provided. By using the method of monotone iterative technique together with upper and lower solutions, conditions for the existence and approximation of minimal and maximal solutions to the BVP under consideration are constructed. Some adequate results for different kinds of Ulam stability are investigated. Maximum error estimates for the corresponding solutions are given as well. Two examples are provided to illustrate the results. KEYWORDS fractional differential equations, monotone iterative technique, Ulam stability, upper and lower solutions extremal solutions Math Meth Appl Sci. 2019;42:969-981.wileyonlinelibrary.com/journal/mma

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