Boundary value problems for a class of stochastic nonlinear fractional order differential equations
OMA-Vol. 4 (2020), Issue 2, pp. 152 – 159 Open Access Full-Text PDF
McSylvester Ejighikeme Omaba, Louis O. Omenyi
Abstract: Consider a class of two-point Boundary Value Problems (BVP) for a stochastic nonlinear fractional order differential equation \(D^\alpha u(t)=\lambda\sqrt{I^\beta[\sigma^2(t,u(t))]}\dot{w}(t),\,\,00\) is a level of the noise term, \(\sigma:[0,1]\times\mathbb{R}\rightarrow\mathbb{R}\) is continuous, \(\dot{w}(t)\) is a generalized derivative of Wiener process (Gaussian white noise), \(D^\alpha\) is the Riemann-Liouville fractional differential operator of order \(\alpha\in (3,4)\) and \(I^\beta,\,\,\beta>0\) is a fractional integral operator. We formulate the solution of the equation via a stochastic Volterra-type equation and investigate its existence and uniqueness under some precise linearity conditions using contraction fixed point theorem. A case of the above BVP for a stochastic nonlinear second order differential equation for \(\alpha=2\) and \(\beta=0\) with \(u(0)=u(1)=0\) is also studied.
Towards understanding the mathematics of the \(2^{nd}\) law of thermodynamics
OMS-Vol. 4 (2020), Issue 1, pp. 466 – 469 Open Access Full-Text PDF
Md. Shafiqul Islam
Abstract: In this paper, the mathematical formulation of \(2^{nd}\) law of thermodynamics has been explained, and the mathematical formulation of \(1^{st}\) law has been revisited from this noble perspective. It is not claimed that the \(2^{nd}\) law of thermodynamics is a redundant of the \(1^{st}\) law, rather I shown here how we can extract the mathematical formulation of the \(2^{nd}\) law from the mathematical formulation of the \(1^{st}\) law of thermodynamics. The Clausius statement of the \(2^{nd}\) law of thermodynamics is, it is impossible to construct a device whose sole effect is the transfer of heat from a cool reservoir to a hot reservoir. An alternative statement of the law is, “All spontaneous processes are irreversible” or, “the entropy of an isolated system always increases”. Having strong experimental evidences, this empirical law is obvious, which tells us the arrow of time and the direction of spontaneous changes.
Measure of noncompactness for nonlinear Hilfer fractional differential equation with nonlocal Riemann–Liouville integral boundary conditions in Banach spaces
OMS-Vol. 4 (2020), Issue 1, pp. 456 – 465 Open Access Full-Text PDF
Abdelatif Boutiara, Maamar Benbachir, Kaddour Guerbati
Abstract: This paper investigates the existence results and uniqueness of solutions for a class of boundary value problems for fractional differential equations with the Hilfer fractional derivative. The reasoning is mainly based upon Mönch’s fixed point theorem associated with the technique of measure of noncompactness. We illustrate our main findings, with a particular case example, included to show the applicability of our outcomes. The boundary conditions introduced in this work are of quite general nature and reduce to many special cases by fixing the parameters involved in the conditions.
Common fixed point results of \(s\)-\(\alpha\) contraction for a pair of maps in \(b\)-dislocated metric spaces
OMA-Vol. 4 (2020), Issue 2, pp. 142 – 151 Open Access Full-Text PDF
Abdissa Fekadu, Kidane Koyas, Solomon Gebregiorgis
Abstract: The purpose of this article is to construct fixed point theorems and prove the existence and uniqueness of common fixed point results of \(s-\alpha\) contraction for a pair of maps in the setting of \(b\) – dislocated metric spaces. Our results extend and generalize several well-known comparable results in the literature. The study procedure we used was that of Zoto and Kumari [1]. Furthermore, we provided an example in support of our main result.
Some integral inequalities for co-ordinated harmonically convex functions via fractional integrals
EASL-Vol. 3 (2020), Issue 4, pp. 60 – 74 Open Access Full-Text PDF
Naila Mehreen, Matloob Anwar
Abstract: In this paper, we find some Hermite-Hadamard type inequalities for co-ordinated harmonically convex functions via fractional integrals.
On the existence of positive solutions of a state-dependent neutral functional differential equation with two state-delay functions
OMA-Vol. 4 (2020), Issue 2, pp. 132 – 141 Open Access Full-Text PDF
El-Sayed, A. M. A, Hamdallah, E. M. A, Ebead, H. R
Abstract: In this paper, we study the existence of positive solutions for an initial value problem of a state-dependent neutral functional differential equation with two state-delay functions. The continuous dependence of the unique solution will be proved. Some especial cases and examples will be given.
