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Öğe BLOW UP AND QUENCHING FOR A PROBLEM WITH NONLINEAR BOUNDARY CONDITIONS(Texas State Univ, 2015) Ozalp, Nuri; Selcuk, BurhanIn this article, we study the blow up behavior of the heat equation u(t) = u(xx) with u(x) (0, t) = u(p) (0,t), u(x) (a,t) = u(q) (a,t). We also study the quenching behavior of the nonlinear parabolic equation v(t) = v(xx) +2v(x)(2) /(1-v) with v(x)(0,t) = (1-v(0, t))(-p+2), v(x)(a,t) = (1-v (a, t)(-q+2). In the blow up problem, if u(0) is a lower solution then we get the blow up occurs in a finite time at the boundary x = a and using positive steady state we give criteria for blow up and non-blow up. In the quenching problem, we show that the only quenching point is x = a and v(t) blows up at the quenching time, under certain conditions and using positive steady state we give criteria for quenching and non-quenching. These analysis is based on the equivalence between the blow up and the quenching for these two equations.Öğe Blow up for non-Newtonian equations with two nonlinear sources(Hacettepe Univ, Fac Sci, 2021) Selcuk, BurhanThis paper studies the following two non-Newtonian equations with nonlinear boundary conditions. Firstly, we show that finite time blow up occurs on the boundary and we get a blow up rate and an estimate for the blow up time of the equation k(t) = (vertical bar k(x)vertical bar(r-2) k(x))(x), (x, t) is an element of (0, L) x (0,T) with k(x) (0,t) = k(alpha) (0, t), k(x) (L,t) = k(beta) (L,t), t is an element of (0, T) and initial function k (x, 0) = k(0) (x), x is an element of [0, L] where r >= 2, alpha, beta and L are positive constants. Secondly, we show that finite time blow up occurs on the boundary, and we get blow up rates and estimates for the blow up time of the equation k(t) = (vertical bar k(x)vertical bar(r-2) k(x))(x) + k(alpha), (x, t) is an element of (0, L) x (0, T) with k(x) (0,t) = 0, k(x) (L,t) = k(beta) (L,t), t is an element of (0,T) and initial function k (x, 0) = k(0) (x), x is an element of [0, L] where r >= 2, alpha, beta and L are positive constants.Öğe Connected Cubic Network Graph(Elsevier - Division Reed Elsevier India Pvt Ltd, 2017) Selcuk, Burhan; Karci, AliHypercube is a popular interconnection network. Due to the popularity of hypercube, more researchers pay a great effort to develop the different variants of hypercube. In this paper, we have proposed a variant of hypercube which is called as Connected Cubic Network Graphs, and have investigated the Hamilton-like properties of Connected Cubic Network Graphs (CCNG). Firstly, we defined CCNG and showed the characteristic analyses of CCNG. Then, we showed that the CCNG has the properties of Hamilton graph, and can be labeled using a Gray coding based recursive algorithm. Finally, we gave the comparison results, a routing algorithm and a bitonic sort algorithm for CCNG. In case of sparsity and cost, CCNG is better than Hypercube. (C) 2017 Karabuk University. Publishing services by Elsevier B.V.Öğe Hamiltonian path, routing, broadcasting algorithms for connected square network graphs(Elsevier - Division Reed Elsevier India Pvt Ltd, 2023) Selcuk, Burhan; Tankuel, Ayse Nur AltintasConnected Square Network Graphs (CSNG) in the study of Selcuk (2022) and Selcuk and Tankul (2022) is reconsidered in this paper. Although (CSNG) is a 2-dimensional mesh structure, the most important feature of this graph is that it is a hypercube variant. For this reason, this study focuses on development algorithms that find solutions to various problems for (CSNG) with the help of hypercube. Firstly, an efficient algorithm that finds the Hamiltonian path is given. Further, two different algorithms that perform the mapping of labels in graph and the unicast routing are given. Furthermore, the parallel process for mapping and unicast routing is discussed. Finally, guidelines are given for broadcasting algorithms. & COPY; 2023 Karabuk University. Publishing services by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Öğe A new hypercube variant: Fractal Cubic Network Graph(Elsevier - Division Reed Elsevier India Pvt Ltd, 2015) Karci, Ali; Selcuk, BurhanHypercube is a popular and more attractive interconnection networks. The attractive properties of hypercube caused the derivation of more variants of hypercube. In this paper, we have proposed two variants of hypercube which was called as Fractal Cubic Network Graphs, and we have investigated the Hamiltonian-like properties of Fractal Cubic Network Graphs FCNG(r)(n). Firstly, Fractal Cubic Network Graphs FCNG(r)(n) are defined by a fractal structure. Further, we show the construction and characteristics analyses of FCNG(r)(n) where r = 1 or r = 2. Therefore, FCNG(r)(n) is a Hamiltonian graph which is obtained by using Gray Code for r = 2 and FCNG(1)(n) is not a Hamiltonian Graph. Furthermore, we have obtained a recursive algorithm which is used to label the nodes of FCNG(2)(n). Finally, we get routing algorithms on FCNG(2)(n) by utilizing routing algorithms on the hypercubes. (C) 2015 Karabuk University. Production and hosting by Elsevier B.V.Öğe On fractal cubic network graphs(Elsevier, 2025-03) Altintas Tankul, Ayse Nur; Selcuk, Burhan; Turan, Muhammed KamilThe fractal cubic network graphs (FCNG), previously studied by Karci and Selcuk (2015), are reviewed in this paper. First, general information about FCNG is provided, and new topological properties of FCNG are presented. Simulations of the topological properties of FCNG, hypercube, and 2D square meshes have been performed, and the results are introduced. Secondly, a strategy for the routing problem for FCNG is presented. A new strategy for the routing path of FCNG is presented and explained with an example, and a recursive algorithm using this strategy is presented. Thirdly, a strategy for the shortest path problem for FCNG with a similar routing strategy is also presented, and a recursive algorithm for this strategy is given. An algorithm for mapping network nodes on a 2D plane and an algorithm for computing the minimum distance connection point between fractals used to construct the shortest path are also provided. These algorithms are illustrated with an example. The running times of the algorithms are also calculated.Öğe The quenching behavior of a nonlinear parabolic equation with a singular boundary condition(Hacettepe Univ, Fac Sci, 2015) Ozalp, Nuri; Selcuk, BurhanIn this paper, we study the quenching behavior of solution of a nonlinear parabolic equation with a singular boundary condition. We prove finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions. Furthermore, we show that the time derivative blows up at quenching point. Also, we get a lower solution and an upper bound for quenching time. Finally, we get a quenching rate and lower bounds for quenching time.Öğe THE QUENCHING BEHAVIOR OF A PARABOLIC SYSTEM(Ankara Univ, Fac Sci, 2013) Selcuk, BurhanIn this paper, we study the quenching behavior of solution of a parabolic system. We prove finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions. Furthermore, we show that the time derivative blows up at quenching time. Finally, we get a quenching criterion by using a comparison lemma and we also get a quenching rate.Öğe THE QUENCHING BEHAVIOR OF A SEMILINEAR HEAT EQUATION WITH A SINGULAR BOUNDARY OUTFLUX(Brown Univ, 2014) Selcuk, Burhan; Ozalp, NuriIn this paper, we study the quenching behavior of the solution of a semilinear heat equation with a singular boundary outflux. We prove a finite-time quenching for the solution. Further, we show that quenching occurs on the boundary under certain conditions and we show that the time derivative blows up at a quenching point. Finally, we get a quenching rate and a lower bound for the quenching time.Öğe Quenching behavior of a semilinear reaction-diffusion system with singular boundary condition(Tubitak Scientific & Technological Research Council Turkey, 2016) Selcuk, BurhanIn this paper, we study the quenching behavior of the solution of a semilinear reaction-diffusion system with singular boundary condition. We first get a local exisence result. Then we prove that the solution quenches only on the right boundary in finite time and the time derivative blows up at the quenching time under certain conditions. Finally, we get lower bounds and upper bounds for quenching time.Öğe QUENCHING BEHAVIOR OF SEMILINEAR HEAT EQUATIONS WITH SINGULAR BOUNDARY CONDITIONS(Texas State Univ, 2015) Selcuk, Burhan; Ozalp, NuriIn this article, we study the quenching behavior of solution to the semilinear heat equation v(t) = v(xx) + f (v), with f(v) = -v(-r) or (1 - v)(-r) and v(x)(0,t) = v(-P)(0,t), v(x)(a,t) = (1-v(a,t))(-q). For this, we utilize the quenching problem u(t) = u(xx) with u(x) (0, t) = u(-P)(0,t), u(x)(a,t) = (1 - u(a,t))(-q). In the second problem, if u(0) is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is x = 0 (x = a) and u(t) blows up at quenching time. Further, we obtain a local solution by using positive steady state. In the first problem, we first obtain a local solution by using monotone iterations. Finally, for f(v) = -v(-r) ((1 - v)(-r)), if v(0) is an upper solution (a lower solution) then we show that quenching occurs in a finite time, the only quenching point is x = 0 (x = a) and v(t) blows up at quenching time.Öğe Quenching estimates for a non-Newtonian filtration equation with singular boundary conditions(Tamkang Univ, 2024) Beauregard, Matthew Alan; Selcuk, BurhanThis study concerns with the quenching features of solutions of the non-Newtonian filtration equation. Various conditions on the initial condition are shown to guarantee quenching at either the left or right boundary. Theoretical quenching rates and lower bounds to the quenching time are determined are certain cases. Numerical experiments are provided to illustrate and provide additional validation of the theoretical predictions to the quenching rates and times.Öğe Quenching for Porous Medium Equations(Tamkang Univ, 2022) Selcuk, BurhanThis paper studies the following two porous medium equations with singular boundary conditions. First, we obtain that finite time quenching on the boundary, as well as k(t) blows up at the same finite time and lower bound estimates of the quenching time of the equation k(t) = (k(n))(xx) + (1 - k)(-alpha), (x, t) is an element of (0, L) x (0, T) with (k(n))(x) (0, t) = 0, (k(n))(x) (L, t) = (1 - k(L, t))(-beta), t is an element of (0, T) and initial function k (x, 0) = k(0) (x), x is an element of[0, L] where n > 1, alpha and beta are positive constants. Second, we obtain that finite time quenching on the boundary, as well as k(t) blows up at the same finite time and a local existence result by the help of steady state of the equation k(t) = (k(n))(xx), (x, t) is an element of (0, L) x (0, T) with (k(n))(x) (0, t) = (1 - k(0, t))(-alpha), (k(n))(x) (L, t) = (1 - k(L, t))(-beta), t is an element of(0, T) and initial function k (x, 0) = k(0) (x), x is an element of [0, L] where n > 1, alpha and beta are positive constants.Öğe QUENCHING PROBLEMS FOR A SEMILINEAR REACTION-DIFFUSION SYSTEM WITH SINGULAR BOUNDARY OUTFLUX(Tamkang Univ, 2016) Selcuk, BurhanIn this paper, we study two quenching problems for the following semilinear reaction-diffusion system: u(t) = u(xx) +(1-v)(-p1), 0 < x < 1, 0 < t < T, v(t) = v(xx) +(1-u)-(p2), 0 < x < 1, 0 < t < T, u(x) (0, t) = 0, u(x) (1, t) = -v (-q1) (1, t), 0 < t < T, v(x) (0, t) = 0, v(x) (1, t) = -u -(q2) (1, t), 0 < t < T, u (x,0) = u(0) (x) < 1, v (x, 0) = v(0) (x) < 1, 0 <= x <= 1, where p(1), p(2), q(1), q(2) are positive constants and u(0)(x), v(0)(x) are positive smooth functions. We firstly get a local exisence result for this system. In the first problem, we show that quenching occurs in finite time, the only quenching point is x = 0 and (u(t), v(t)) blows up at the quenching time under the certain conditions. In the second problem, we show that quenching occurs in finite time, the only quenching point is x = 1 and (u(t), v(t)) blows up at the quenching time under the certain conditions.Öğe Using gauss - Jordan elimination method with CUDA for linear circuit equation systems(Elsevier Science Bv, 2012) Atasoy, Nesrin Aydin; Sen, Baha; Selcuk, BurhanMany scientific and engineering problems can use a system of linear equations. In this study, solution of Linear Circuit Equation System (LCES) for an nxn matrix using Compute Unified Device Architecture (CUDA) is described. Solution of LCES is realized on Graphics Processing Unit (GPU) instead of Central Processing Unit (CPU). CUDA is a parallel computing architecture developed by NVIDIA. Linear Circuits include resistance, impedance, capacitance, dependent - independent current sources and DC, AC voltage source. In this study, solutions of circuits that include resistance, independent current sources and DC voltage source have analyzes. Circuit analysis frequently involves solution of linear simultaneous equations that are solved Gauss-Jordan Elimination Method in this study. Gauss-Jordan Elimination is a variant of Gaussian Elimination that a method of solving a linear system equations (Ax=B). Gauss-Jordan Elimination is an algorithm for getting matrices in reduced row echelon form using elementary row operations. Gaussian Elimination has two parts. The first part (Forward Elimination) reduces a given system to triangular form. The second step uses back substitution to find the solution of the triangular echelon form system Because of elements of unknowns column matrix are dependent on each other, second step algorithm is not appropriate for parallel programming. Two parts of Gauss-Jordan Elimination are not like Gaussian Elimination's part so it is preferred. GPU implementation is more faster than solution of linear equation systems on CPU. (C) 2011 Published by Elsevier Ltd.
