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Öğe Bükeyliği Olmayan Durum Fonksiyonlu Birinci Basamaktan Denklem İçin Riemann Problemi(Beykent Üniversitesi, 2014) Ulas, Ozgur; Rasulov, MahirMakalede, bükeyliği olmayan durum fonksiyonlu birinci basamaktan nonlineer hiperbolik tür denklem için Riemann probleminin gerçek çözümleri elde edilmiştir. Bunun için bazı durumda esas probleme bilinen anlamda denk olan ve özel yolla kurulmuş yardımcı problem dahil edilmiştir. Bazı durumlarda ise problemin karakterine uygun olarak aşağı veya yukarı konveks katmanlar kurulmuş ve bunların yardımıyla gerçek çözümler bulunmuştur.Öğe The D'Alembert Type Solution of the Cauchy Problem for the Homogeneous with Respect to Fourth Order Derivatives for Hyperbolic Equation(Springer International Publishing Ag, 2017) Rasulov, Mahir; Aslan, Zafer; Sinsoysal, Bahaddin; Bal, HakanIn this paper the D'Alembert type solution for the following Sigma(i=0) a(i) partial derivative(4)u (x, t)/partial derivative t(4)-(i)partial derivative x(i) = 0, partial derivative(k) u(x, t)/partial derivative t(k) |(t=0) = phi(k) (x), (k = 0, 1, 2, 3) Cauchy problem is constructed. Here, a(i), (i = 1, 2, 3, 4) and fk (x), (k = 1, 2, 3, 4) are given constants and functions, respectively. The cases where the roots of the characteristic equation are folded are examined and compact expressions for the solutions are obtained. The obtained solutions allow proving the uniqueness and existence of the solutions of the considered problem.Öğe Diferansiyel Denklemler Sisteminin Rezidü Çözümü ve Sabit Gerilim Kaynaklı RC Devresi Problemine Uygulanması(Beykent Üniversitesi, 2013) Sinsoysal, Bahaddin; Rasulov, MahirMakalede sabit katsayılı adi diferansiyel denklemler sistemi için yazılmış Cauchy probleminin rezidü metodu ile gerçek çözümü elde edilmiş ve söz konusu metot uygulanarak sabit gerilimli bir RC devre probleminin çözümünün bulunması için uygulanmıştır.Öğe Efficient Numerical Method of the 1D Motion of Two-Phase Fluid through Porous Medium in a Class of Discontinuous Functions(Springer-Verlag Berlin, 2009) Sinsoysal, Bahaddin; Rasulov, MahirIn this paper a new numerical method for nonlinear system of partial differential equations which describes motion of a two-phase compressible fluid in microscopic porous medium in a class of discontinuous functions is suggested. For this aim, at first, the system of equations is split in two equations according to physical parameters. First one with respect to water saturation, and the other one with respect to pressure. Then, the special auxiliary problem having some advantages over the main problem is introduced. This auxiliary problem permits to develop the efficient numerical algorithm for obtaining the solution. Using the solution of the auxiliary problem, the solution of the main problem, which expresses all the physical properties is found.Öğe Finite Differences Method for the First Order 2-D Partial Equation(Springer-Verlag Berlin, 2013) Rasulov, Mahir; Sahin, E. Ilhan; Soguksu, M. GokhanIn this study a new method for finding exact solution of the Cauchy problem subject to a discontinuous initial profile for the two dimensional scalar conservation laws is suggested. For this aim, first, some properties of the weak solution of the linearized equation are investigated. Taking these properties into consideration an auxiliary problem having some advantages over the main problem is introduced. The proposed auxiliary problems also permit us to develop effective different numerical algorithms for finding the solutions. Some computer experiments are carried out.Öğe The Finite Differences Schemes For First Order Nonlinear System Equations In A Class Of Discontinuous Functions(Beykent Üniversitesi, 2007) Rasulov, Mahir; Sinsoysal, BahaddinIn this paper a new finite differences schemes of the Cauchy and initialboundary value problem, for the first order system differential equations which describe some conservation laws is suggested. At first an auxiliary problem which is equivalent to the main problem in certain sense is introduced. On the basis of the auxiliary problem the simple and economical algorithms from computational point of view are proposed.Öğe Grid Method for Solving the Flow of Traffic Problem on the Highway in a Class of Discontinuous Functions(Springer-Verlag Berlin, 2009) Rasulov, Mahir; Cocer, KenanIn this paper a new method for obtaining an exact and numerical solution of initial value problem for a first order nonlinear partial equation which describes the traffic flow on highway in a class of discontinuous functions is developed. For this goal, at first, the special auxiliary problem having some advantages over the main problem has been suggested and using the solution of the auxiliary problem an original method for finding the weak solution of the main problem has been suggested. Using the advantages of the auxiliary problem the new numerical method for obtaining a solution which expresses the all physical properties accurately is purposed, too.Öğe Residue Method for The Solution of Heat Equation with Nonlocal Boundary Condition(Beykent Üniversitesi, 2008) Rasulov, Mahir; Sinsoysal, BahaddinIn this paper by using the residue method, the exact solution of mixed problem for linear heat equation with nonlocal boundary conditions is obtained. For this aim, at first, the formula for expansion of an arbitrary function in a series of residues of the solution of the corresponding spectral problem is proved. Further, this formula is used to show that if the mixed boundary-value problem in question has a sufficiently smooth solution, this solution can be represented by the given residue formula.Öğe Riemann Problem for First-Order Partial Equations Without the Convexity of a State Functions(Springer-Verlag Berlin, 2015) Rasulov, Mahir; Ulas, S. OzgurIn this work, the exact solution of the Riemann problem for first-order nonlinear partial equation with non-convex state function in Q(T) = {(x, t)vertical bar x is an element of I = (-infinity, infinity), t is an element of [0, T)} subset of R-2 is found. Here F is an element of C-2 (Q(T)) and F '' (u) change their signs, that is F(u) has convex and concave parts. In particular, the state function F (u) = -cos u on [pi/2, 3 pi/2] and [pi/2, 5 pi/2] is discussed. For this, when it is necessary, the auxiliary problem which is equivalent to the main problem is introduced. The solution of the proposed problem permits constructing the weak solution of the main problem that conserves the entropy condition. In some cases, depending on the nature of the investigated problem a convex or a concave hull is constructed. Thus, the exact solutions are found by using these functions.Öğe The Study of Filtration of Two Phase Fluid in a Porous Medium in a Class of Discontinuous Functions(Springer-Verlag Berlin, 2009) Rasulov, Mahir; Kul, R. HalukIn this study a new numerical method for obtaining the solution of the problem for 2-D system of equations which describes the filtration of the multi-phase fluid in porous medium with suitable initial and boundary conditions is proposed. For this aim, the special auxiliary problem having some advantages over the main problem is introduced. An efficient and accurate numerical algorithm based on the auxiliary problem is derived. Furthermore, some results of numerical experiments from the related subjects of physics are presented.
