arXiv subjects

Fatma Zürnacı-Yetiş

Publications and source records attributed to Fatma Zürnacı-Yetiş.

4 recordsLinked to original sources

arXiv2026

Fully discrete scheme for the fifth-order KdV-Burgers-Fisher equation using Strang splitting and Fourier collocation methods

Operator splitting is an effective technique for the numerical solution of nonlinear partial differential equations by decomposing a complex problem into simpler subproblems. In this study, we present and analyze a fully discrete scheme for the fifth-order Korteweg-de Vries-Burgers-Fisher equation (KBF) by combining Strang splitting for time discretization with the Fourier collocation method for spatial discretization. In particular, the Fourier collocation method is an essential component of the proposed fully discrete scheme and yields spectral accuracy in space under suitable regularity assumptions. The KBF equation describes the interaction of reaction, dissipative, and dispersive mechanisms by incorporating the Fisher reaction term together with Burgers-type diffusion and higher-order KdV dispersion. The equation is split into a linear operator and a nonlinear operator, and the resulting subproblems are solved within the Strang splitting framework. Convergence is analyzed in the Sobolev space $H^s$. The local error is derived using operator-theoretic arguments in Banach spaces together with Lie commutator estimates, while the global error is obtained using the Lady Windermere's fan argument. The analysis yields second-order convergence in time and spectral convergence in space. Numerical results confirm the theoretical error estimates and demonstrate the accuracy of the proposed fully discrete scheme.

math.NA↗

arXiv2026

$h$-$γ$ Blossoming, $h$-$γ$ Bernstein Bases, and $h$-$γ$ Bézier Curves for Translation Invariant $\left(γ_{1},γ_{2}\right)$ Spaces

A $\left(γ_{1}, γ_{2}\right)$ space of order $n$ is a space of univariate functions spanned by $\left\{γ_{1}^{n-k}(x), γ_{2}^{k}(x)\right\}_{k=0}^{n}$. A $\left(γ_{1}, γ_{2}\right)$ space is said to be translation invariant if $γ_{1}(x-h)$ and $γ_{2}(x-h)$ can be expressed as nonsingular linear combinations of $γ_{1}(x)$ and $γ_{2}(x)$. Translation invariant $\left(γ_{1}, γ_{2}\right)$ spaces include polynomials $\left(γ_{1}(x)=1, γ_{2}(x)=x\right)$, trigonometric functions $\left(γ_{1}(x)=\cos x, γ_{2}(x)=\sin x\right)$, hyperbolic functions $\left(γ_{1}(x)=\cosh x, γ_{2}(x)=\sinh x\right)$, and their discrete analogues. We merge $γ$-blossoming for $\left(γ_{1}, γ_{2}\right)$ spaces with $h$-blossoming for $h$-Bernstein bases and $h$-Bézier curves to construct a novel $h$-$γ$ blossom for translation invariant $\left(γ_{1}, γ_{2}\right)$ spaces generated by two continuous, linearly independent functions $γ_{1}$ and $γ_{2}$. Based on this $h$-$γ$ blossom, we define $h$-$γ$ Bernstein bases and $h$-$γ$ Bézier curves and study their properties. We derive recursive evaluation algorithms, subdivision procedures, Marsden identities, and formulas for degree elevation and interpolation for these $h$-$γ$ Bernstein and $h$-$γ$ Bézier schemes.

math.NA↗

arXiv2026

Non-polynomial divided difference and blossoming

Two notable examples of dual functionals in approximation theory and computer-aided geometric design are the blossom and the divided difference operator. Both of these dual functionals satisfy a similar set of formulas and identities. Moreover, the divided differences of polynomials can be expressed in terms of the blossom. In this paper, an extended non-polynomial homogeneous blossom for a wide collection of spline spaces, including trigonometric splines, hyperbolic splines, and special Müntz spaces of splines, is defined. It is shown that there is a relation between the non-polynomial divided difference and the blossom, which is analogous to the polynomial case.

math.NA↗

arXiv2025

$h$-Trigonometric B-splines

We introduce discrete analogues of the exponential, sine, and cosine functions. Then using a discrete trigonometric version of a non-polynomial divided difference, we define discrete analogues of the trigonometric B-splines. We derive a two-term recurrence relation, a two-term formula for the discrete derivative, and two variants of the Marsden identity for these discrete trigonometric B-splines. Since the classical exponential, sine, and cosine functions are limiting cases of their discrete analogues, we conclude that many of the standard results for classical polynomial B-splines extend naturally both to trigonometric B-splines and to discrete trigonometric B-splines.

math.NA↗