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We consider the nonlinear Sturm-Liouville problem −(p(x)u′(x))′ + q(x)u(x) = f(x, u(x)) + h(x), in (0, π), c00u(0) + c01u ′(0) = 0, c10u(π) + c11u ′(π) = 0, where: p ∈ C1[0, π], q ∈ C0[0, π], with p(x) > 0 for all x ∈ [0, π]; ci0 + ci1 > 0, i = 0, 1; h ∈ L2(0, π). We suppose that f : [0, π] × R → R is continuous and there exist increasing functions ζl, ζu : [0,∞)→ R, and positive constants A, B...

We consider the nonlinear Sturm-Liouville problem −(p(x)u′(x))′ + q(x)u(x) = f(x, u(x), u′(x)), in (0, π), (1) c00u(0) + c01u ′(0) = 0, c10u(π) + c11u ′(π) = 0, (2) where p ∈ C1[0, π], q ∈ C0[0, π], with p(x) > 0, x ∈ [0, π], and ci0 + ci1 > 0, i = 0, 1. We suppose that f : [0, π] × R2 → R is continuous and there exist increasing functions ζl, ζu : [0,∞) → R, and a constant B, such that limt→∞ ...

The paper presents a sixth-order numerical algorithm for studying the completely integrable CamassHolm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We...

We present an explicit sixth order compact finite difference scheme for fast high accuracy numerical solutions of the two dimensional convection diffusion equation with variable coefficients. The sixth order scheme is based on the well-known fourth order compact scheme, the Richardson extrapolation technique, and an operator interpolation scheme. For a particular implementation, we use multisca...