arXiv subjects

Victor Nijimbere

Publications and source records attributed to Victor Nijimbere.

8 recordsLinked to original sources

arXiv2020

Analytical valuation of some non-elementary integrals involving some exponential, hyperbolic and trigonometric elementary functions and derivation of new probability measures generalizing the gamma-type and normal distributions

The non-elementary integrals involving elementary exponential, hyperbolic and trigonometric functions, $ \int x^αe^{ηx^β}dx, \int x^α\cosh\left(ηx^β\right)dx, \int x^α\sinh\left(ηx^β\right)dx, \int x^α\cos\left(ηx^β\right)dx$ and $\int x^α\sin\left(ηx^β\right)dx $ where $α, η$ and $β$ are real or complex constants are evaluated in terms of the confluent hypergeometric function $_1F_1$ and the hypergeometric function $_1F_2$. The hyperbolic and Euler identities are used to derive some identities involving exponential, hyperbolic, trigonometric functions and the hypergeometric functions $_1F_1$ and $_1F_2$. Having evaluated, these non-elementary integrals, some new probability measures generalizing the gamma-type and normal distributions are also obtained. The obtained generalized distributions may, for example, allow to perform better statistical tests than those already known (e.g. chi-square ($χ^2$) statistical tests and those based on central limit theorem (CLT)).

math.GM↗

arXiv2020

Evaluation of some non-elementary integrals involving the generalized hypergeometric function with some applications

The indefinite integral $$ \int x^αe^{ηx^β}\,_pF_q (a_1, a_2, \cdot\cdot\cdot a_p; b_1, b_2, \cdot\cdot\cdot, b_q; λx^γ)dx, $$ where $α, η, β, λ, γ\ne0$ are real or complex constants and $_pF_q$ is the generalized hypergeometric function, is evaluated in terms of an infinite series involving the generalized hypergeometric function. Related integrals in which the exponential function $e^{ηx^β}$ is either replaced by the hyperbolic function $\cosh\left(ηx^β\right)$ or $\sinh\left(ηx^β\right)$, or the sinusoidal function $\cos\left(ηx^β\right)$ or $\sin\left(ηx^β\right)$, are also evaluated in terms of infinite series involving the generalized hypergeometric function $_pF_q$. Some application examples from applied analysis, in which some new Fourier and Laplace integrals (or transforms) are evaluated, are given. The analytical solution of the Orr-Sommerfeld equation (with a linear mean flow background) in the short-wave limit is expressed in terms of some infinite series involving the hypergeometric series $_2F_3$. Making use of the hyperbolic and Euler identities, some interesting series identities involving exponential, hyperbolic, trigonometric functions and the generalized hypergeometric function are also derived.

math.CA↗

arXiv2019

Implementation of a Wiener Chaos Expansion Method for the Numerical Solution of the Stochastic Generalized Kuramoto-Sivashinsky Equation driven by Brownian motion forcing

Numerical computations based on the Wiener Chaos Expansion (WCE) are carried out to approximate the solutions of the stochastic generalized Kuramoto--Sivashinsky (SgKS) equation driven by Brownian motion forcing. In the assessment of the accuracy of the WCE based approximate numerical solutions, the WCE based solutions are contrasted with semi-analytical solutions, and the absolute and relative errors are evaluated. It is found that the absolute error is $O(ςt)$, where $ς$ is small constant and $t$ is the time variabe; and the relative error is order $10^{-2}$ or less. This demonstrates that numerical methods based on the WCE are powerful tools to solve the SgKS equation or other related stochastic evolution equations.

math.NA↗

arXiv2019

Asymptotic approximation of the eigenvalues and the eigenfunctions for the Orr-Sommerfeld equation on infinite intervals

Asymptotic eigenvalues and eigenfunctions for the Orr-Sommerfeld equation in two and three dimensional incompressible flows on an infinite domain and on a semi-infinite domain are obtained. Two configurations are considered, one in which a short-wave limit approximation is used, and another in which a long-wave limit approximation is used. In the short-wave limit, WKB methods are utilized to estimate the eigenvalues, and the eigenfunctions are approximated in terms Green's functions. The procedure consists of transforming the Orr-Sommerfeld equation into a system of two second order ordinary differential equations for which eigenvalues and eigenfunctions can be approximated. The approximated eigenvalues can, for instance, be used as a starting point in predicting transitions in boundary layers with computer simulations (computational fluid dynamics). In the long-wave limit approximation, solutions are expressed in terms of generalized hypergeometric functions.

math.AP↗

arXiv2018

Evaluation of the non-elementary integral $\int e^{λx^α} dx, α\ge2$, and other related integrals

A formula for the non-elementary integral $\int e^{λx^α} dx$ where $α$ is real and greater or equal two, is obtained in terms of the confluent hypergeometric function $_1F_1$. This result is verified by directly evaluating the area under the Gaussian Bell curve, corresponding to $α= 2$, using the asymptotic expression for the confluent hypergeometric function and the Fundamental Theorem of Calculus (FTC). Two different but equivalent expressions, one in terms of the confluent hypergeometric function $_1F_1$ and another one in terms of the hypergeometric function $_1F_2$, are obtained for each of these integrals, $\int \cosh(λx^α)dx$, $\int \sinh(λx^α)dx$, $\int \cos(λx^α)dx$ and $\int \sin(λx^α)dx$, $λ\in \mathbb{C}, α\ge2$. And the hypergeometric function $_1F_2$ is expressed in terms of the confluent hypergeometric function $_1F_1$. Some of the applications of the non-elementary integral $\int e^{λx^α}dx,α\ge2$ such as the Gaussian distribution and the Maxwell-Bortsman distribution are given.

math.CA↗

arXiv2018

Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: Part I

The non-elementary integrals $\text{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α\leβ+1$ and $\text{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α\le2β+1$, where $\{β,α\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric functions $_{1}F_2$ and $_{2}F_3$, and their asymptotic expressions for $|x|\gg1$ are also derived. The integrals of the form $\int [\sin^n{(λx^β)}/(λx^α)] dx$ and $\int [\cos^n{(λx^β)}/(λx^α)] dx$, where $n$ is a positive integer, are expressed in terms $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$, and then evaluated. $\text{Si}_{β,α}$ and $\text{Ci}_{β,α}$ are also evaluated in terms of the hypergeometric function $_{2}F_2$. And so, the hypergeometric functions, $_{1}F_2$ and $_{2}F_3$, are expressed in terms of $_{2}F_2$.The exponential integral $\text{Ei}_{β,α}=\int (e^{λx^β}/x^α) dx$ where $β\ge1$ and $α\leβ+1$ and the logarithmic integral $\text{Li}=\int_μ^{x} dt/\ln{t}, μ>1$ are also expressed in terms of $_{2}F_2$, and their asymptotic expressions are investigated. It is found that for $x\ggμ$, $\text{Li}\sim {x}/{\ln{x}}+\ln{\left(\frac{\ln{x}}{\lnμ}\right)}-2-\lnμ\hspace{.075cm} _{2}F_{2}(1,1;2,2;\lnμ)$, where the term $\ln{\left(\frac{\ln{x}}{\lnμ}\right)}-2-\lnμ\hspace{.075cm} _{2}F_{2}(1,1;2,2;\lnμ)$ is added to the known expression in mathematical literature $\text{Li}\sim {x}/{\ln{x}}$.

math.CA↗

arXiv2018

Evaluation of some non-elementary integrals involving sine, cosine, exponential and logarithmic integrals: Part II

The non-elementary integrals $\mbox{Si}_{β,α}=\int [\sin{(λx^β)}/(λx^α)] dx,β\ge1,α>β+1$ and $\mbox{Ci}_{β,α}=\int [\cos{(λx^β)}/(λx^α)] dx, β\ge1, α>2β+1$, where $\{β,α\}\in\mathbb{R}$, are evaluated in terms of the hypergeometric function $_{2}F_3$. On the other hand, the exponential integral $\mbox{Ei}_{β,α}=\int (e^{λx^β}/x^α) dx, β\ge1, α>β+1$ is expressed in terms of $_{2}F_2$. The method used to evaluate these integrals consists of expanding the integrand as a Taylor series and integrating the series term by term.

math.CA↗

arXiv2017

Analytical evaluation and asymptotic evaluation of Dawson's integral and related functions in mathematical physics

Dawson's integral and related functions in mathematical physics that include the complex error function (Faddeeva's integral), Fried-Conte (plasma dispersion) function, (Jackson) function, Fresnel function and Gordeyev's integral are analytically evaluated in terms of the confluent hypergeometric function.And hence, the asymptotic expansions of these functions on the complex plane $\mathbb{C}$ are derived using the asymptotic expansion of the confluent hypergeometric function.

math.CA↗