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Ibrahim O. Sarumi

Publications and source records attributed to Ibrahim O. Sarumi.

5 recordsLinked to original sources
arXiv2026

Analysis and Efficient Sylvester-Based Implementation of a Dimension-Split ETD2RK Scheme for Multidimensional Reaction-Diffusion Equations

We propose and analyze a second-order, dimension-split exponential time differencing Runge--Kutta scheme (ETD2RK-DS) for multidimensional reaction--diffusion equations in two and three spatial dimensions. Under mild assumptions on the nonlinear source term, we establish uniform stability bounds and prove second-order temporal convergence for the underlying dimension-split scheme. To enable efficient implementation, we employ Padé approximations of the matrix exponential, converting each required matrix-exponential--vector product into the solution of a shifted linear system. A convergence analysis of the resulting Padé-based ETD2RK-DS formulation is provided. We derive explicit and reproducible tensor-slicing and reshaping algorithms that realize the dimension-splitting strategy, decomposing multidimensional systems into collections of independent one-dimensional problems. This leads to a reduction of the dominant per-time-step computational cost from $\mathcal{O}(m^3)$ to $\mathcal{O}(m^2)$ in two dimensions and from $\mathcal{O}(m^5)$ to $\mathcal{O}(m^3)$ in three dimensions when compared with banded LU solvers for the unsplit problem, where $m$ denotes the number of grid points per spatial direction. Furthermore, we develop a Sylvester-equation reformulation of the resulting one-dimensional systems, enabling a highly efficient spectral implementation based on reusable eigendecompositions, matrix--vector multiplications, and Hadamard divisions. Numerical experiments in two and three dimensions, including a coupled FitzHugh--Nagumo system, confirm the second-order temporal accuracy, stability of the underlying scheme, and scalability of the proposed ETD2RK-DS framework, as well as the substantial computational advantages of the Sylvester-based implementation over classical LU-based solvers.

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arXiv2023

Rational Approximations for Oscillatory Two-Parameter Mittag-Leffler Function

The two-parameter Mittag-Leffler function $E_{α, β}$ is of fundamental importance in fractional calculus. It appears frequently in the solutions of fractional differential and integral equations. Nonetheless, this vital function is often expensive to compute. Several attempts have been made to construct cost-effective and accurate approximations. These attempts focus mainly on the completely monotone Mittag-Leffler functions. However, when $α> 1$ the monotonicity property is largely lost and as such roots and oscillations are exhibited. Consequently, existing approximants constructed mainly for $α\in (0,1)$ often fail to capture this oscillatory behavior. In this paper, we construct computationally efficient and accurate rational approximants for $E_{α, β}(-t)$, $t \ge 0$, with $α\in (1,2)$. This construction is fundamentally based on the decomposition of Mittag-Leffler function with real roots into one without and a polynomial. Following which new approximants are constructed by combining the global Padé approximation with a polynomial of appropriate degree. The rational approximants are extended to approximation of matrix Mittag-Leffler and different approaches to achieve efficient implementation for matrix arguments are discussed. Numerical experiments are provided to illustrate the significant accuracy improvement achieved by the proposed approximants.

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arXiv2019

Highly Accurate Global Padé Approximations of Generalized Mittag-Leffler Function and its Inverse

The two-parametric Mittag-Leffler function (MLF), $E_{α,β}$, is fundamental to the study and simulation of fractional differential and integral equations. However, these functions are computationally expensive and their numerical implementations are challenging. In this paper, we present a unified framework for developing global rational approximants of $E_{α,β}(-x)$, $x>0$, with $\{ (α,β): 0 < α\leq 1, β\geq α, (α,β) \ne (1,1) \}$. This framework is based on the series definition and the asymptotic expansion at infinity. In particular, we develop three types of fourth-order global rational approximations and discuss how they could be used to approximate the inverse function. Unlike existing approximations which are either limited to MLF of one parameter or of low accuracy for the two-parametric MLF, our rational approximants are of fourth order accuracy and have low percentage error globally. For efficient utilization, we study the partial fraction decomposition and use them to approximate the two-parametric MLF with a matrix argument which arise in the solutions of fractional evolution differential and integral equations.

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arXiv2019

Inverse source in two-parameter anomalous diffusion, numerical algorithms and simulations over graded time-meshes

We consider an inverse source two-parameter sub-diffusion model subject to a nonlocal initial condition. The problem models several physical processes, among them are the microwave heating and light propagation in photoelectric cells. A bi-orthogonal pair of bases is employed to construct a series representation of the solution and a Volterra integral equation for the source term. We develop a numerical algorithm for approximating the unknown time-dependent source term. Due to the singularity of the solution near $t=0$, a graded mesh is used to improve the convergence rate. Numerical experiments are provided to illustrate the expected analytical order of convergence.

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