A unified integral operator and further its consequences

OMA-Vol. 4 (2020), Issue 1, pp. 1 – 7 Open Access Full-Text PDF
Ghulam Farid
Abstract: The aim of this paper is to construct left sided and right sided integral operators in a unified form. These integral operators produce various well known integral operators in the theory of fractional calculus. Formulated integral operators of this study include generalized fractional integral operators of Riemann-Liouville type and operators containing Mittag-Leffler functions in their kernels. Also boundedness of all these fractional integral operators is derived from the boundedness of unified integral operators. The existence of new integral operators may have useful consequences in applied sciences besides in fractional calculus.
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Positive solutions for boundary value problem of sixth-order elastic beam equation

OMS-Vol. 4 (2020), Issue 1, pp. 9 – 17 Open Access Full-Text PDF
Zouaoui Bekri, Slimane Benaicha
Abstract: In this paper, we study the existence of positive solutions for boundary value problem of sixth-order elastic beam equation of the form \(-u^{(6)}(t)=q(t)f(t,u(t),u^{‘}(t),u^{”}(t),u^{”’}(t),u^{(4)}(t),u^{(5)}(t)),~~0<t<1,\) with conditions \(u(0)=u^{‘}(1)=u^{”}(0)=u^{”’}(1)=u^{(4)}(0)=u^{(5)}(1)=0,\) where \(f\in C([0,1]\times[0,\infty)\times[0,\infty)\times(-\infty,0]\times(-\infty,0]\times[0,\infty)\times[0,\infty)\rightarrow [0,\infty))\). The boundary conditions describe the deformation of an elastic beam simply supported at left and clamped at right by sliding clamps. We give sufficient conditions that allow us to obtain the existence of positive solution. The main tool used in the proof is the Leray-Schauder nonlinear alternative and Leray-Schauder fixed point theorem. As an application, we also give example to illustrate the results obtained.
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A fixed point theorem for generalized weakly contractive mappings in \(b\)-metric spaces

OMS-Vol. 4 (2020), Issue 1, pp. 1 – 8 Open Access Full-Text PDF
Eliyas Zinab, Kidane Koyas, Aynalem Girma
Abstract: In this paper we establish a fixed point theorem for generalized weakly contractive mappings in the setting of \(b\)-metric spaces and prove the existence and uniqueness of a fixed point for a self-mappings satisfying the established theorem. Our result extends and generalizes the result of Cho [1]. Finally, we provided an example in the support of our main result.
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Existence of solution for a nonlinear fifth-order three-point boundary value problem

OMA-Vol. 3 (2019), Issue 2, pp. 125 – 136 Open Access Full-Text PDF
Zouaoui Bekri, Slimane Benaicha
Abstract: In this paper, we explore the existence of nontrivial solution for the fifth-order three-point boundary value problem of the form \(u^{(5)}(t)+f(t,u(t))=0,\quad\text 0<t<1,\) with boundary conditions \(u(0)=0,\quad u^{‘}(0)=u^{”}(0)=u^{”’}(0)=0,\quad u(1)=\alpha u(\eta),\) where \(0<\eta<1\), \(\alpha\in\mathbb{R}\), \(\alpha\eta^{4}\neq1\), \(f\in C([0,1]\times\mathbb{R},\mathbb{R})\). Under certain growth conditions on the non-linearity \(f\) and using Leray-Schauder nonlinear alternative, we prove the existence of at least one solution of the posed problem. Furthermore, the obtained results are further illustrated by mean of some examples.
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Existence and uniqueness results for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations

OMA-Vol. 3 (2019), Issue 2, pp. 106 – 111 Open Access Full-Text PDF
Abdelouaheb Ardjouni, Adel Lachouri, Ahcene Djoudi
Abstract: In this paper, we use the Banach fixed point theorem to obtain the existence, interval of existence and uniqueness of solutions for nonlinear hybrid implicit Caputo-Hadamard fractional differential equations. We also use the generalization of Gronwall’s inequality to show the estimate of the solutions.
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New inequalities based on harmonic log-convex functions

OMA-Vol. 3 (2019), Issue 2, pp. 103 – 105 Open Access Full-Text PDF
Imran Abbas Baloch, Silvestru Sever Dragomir
Abstract: Harmonic convexity is very important new class of non-convex functions, it gained prominence in the Theory of Inequalities and Applications as well as in the rest of Mathematics’s branches. The harmonic convexity of a function is the basis for many inequalities in mathematics. Furthermore, harmonic convexity provides an analytic tool to estimate several known definite integrals like \(\int_{a}^{b} \frac{e^{x}}{x^{n}}dx\), \(\int_{a}^{b} e^{x^{2}} dx\), \(\int_{a}^{b} \frac{\sin x}{x^{n}}dx\) and \(\int_{a}^{b} \frac{\cos x}{x^{n}}dx\) \(\forall n \in \mathbb{N}\), where \(a,b \in (0,\infty)\). In this article, some un-weighted inequalities of Hermite-Hadamard type for harmonic log-convex functions defined on real intervals are given.
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The analysis of failure and reliability factors of impressed current cathodic protection (ICCP) design toward underwater line of warship (Case study in kri kcr-40 type)

EASL-Vol. 2 (2019), Issue 4, pp. 45 – 56 Open Access Full-Text PDF
Nengah Putra, Romie Oktovianus Bura, Sovian Aritonang, Djoko Navalino, Joni Widjayanto
Abstract: Corrosion at the bottom of a ship’s water line can result in personnel and material safety risks. There are 2 (two) ways to protect against corrosion, they are passive protection (by painting) and active protection (by cathodic protection method). In the KRI with KCR-40 type, the design of the bottom line of the ship’s waterline protection has been carried out with ICCP, but the value of its failure risk and reliability is unknown, both functional and designs, so that the design of the tool cannot be used maximally. This research aimed to determine the factors of failure and reliability value of the design-based ICCP (Reliability by Design) with the FTA and FMEA approach, the FTA aimed to identify the risks that contribute to the failure. The main factors causing failure in the design of ICCP tools occur in the component of Steel potential indicator and rectifier indicator with a failure mode not pointing to the correct number, this will result in corrosion control which is expected to be uncontrolled properly and correctly due to incorrect data input. After analyzing the FTA, the reliability value was 33%. Mitigation of tool components that have a high level of risk among other things in the indicator of steel potential and rectifier indicators: the first was to redesign the laying of some components of the tool compilers to pay attention to the circulating circulation in the box so that the tool works more optimally, the second was to carry out periodic control while the device was operating, and third was to ensure that the electrical power used was stable so there were no problems with the ICCP device while the ICCP device was operating.
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Generalized dynamics on the penetration of two phase fuel spray using differential transform method

EASL-Vol. 2 (2019), Issue 4, pp. 33 – 44 Open Access Full-Text PDF
B. Y. Ogunmola, A. A. Yinusa
Abstract: The penetration of fuel spray as a result of the mixture of fuel droplet and entrained air usually generate nonlinear models whose solutions are normally difficult to realize analytically. This present study presents general approximate analytical solution to such problem by employing Differential transform Method (DTM). At the level of two-phase flow, the spray droplets and the entrained air have the same flow velocity. In order to fully understand the process, the parameters present in the governing equations are carefully studied. The obtained solution employing DTM is verified with Numerical Runge-Kutta (RKF45) and also compared with similar past works. Furthermore, the acquired results for different ambient densities and injection velocities are depicted and discussed. The results illustrate that continuous increase in the initial velocity and orifice diameter cause a corresponding increase in spray penetration while an antonymous effect is noticed for an increased semi cone angle and density. This work will find vital applications in the optimization of systems whose operation are influence by the aforementioned spray penetration processes.
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Pseudo-valuations and pseudo-metric on JU-algebras

OMS-Vol. 3 (2019), Issue 1, pp. 440 – 446 Open Access Full-Text PDF
Usman Ali, Moin A. Ansari, Masood Ur Rehman
Abstract: In this paper we have introduced the concept of pseudo-valuations on JU-algebras and have investigated the relationship between pseudo-valuations and ideals of JU-algebras. Conditions for a real-valued function to be a pseudo-valuation on JU-algebras are given and results based on them have been shown. We have also defined and studied pseudo-metric on JU-algebras and have proved that \(\vartheta\) being a valuation on a JU-algebras \(A\), the operation \(\diamond\) in \(A\) is uniformly continuous.
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