Abstract:In this paper, we study the positive properties of the Green function for the following two-term fractional differential equation0+ is the standard Riemann-Liouville derivative. As an application, the existence and uniqueness of positive solution are obtained under the singular conditions. Moreover, an iterative scheme is established to approximate the unique positive solution.
“…where c 11 = −3 4 10 + 4a 2 10 − 2b, c 12 = c 18 = 12 4 10 z 1 − 16a 2 10 z 1 − 12 3 10 11 + 12 3 10 12 + 8a 10 11 − 8a 10 12 + 8bz 1 , c 13 = c 17 = −18 4 10 z 2 1 + 24a 2 10 z 2 1 + 48 3 10 11 z 1 − 36 3 10 12 z 1 − 32a 10 11 z 1 + 24a 10 From (7), we obtain the indeterminate relations to elliptic solutions of Equation PIV with pole at z = 0:…”
Section: Resultsmentioning
confidence: 99%
“…A great deal of research has been conducted worldwide in the study of nonlinear differential equations. [1][2][3][4][5][6][7][8][9][10] The 6 Painlevé equations have been identified as the most important nonlinear ordinary differential equations ODEs. 11 In the past 2 decades, there is continuing interest in solutions of Painlevé equations.…”
In this paper, the complex method is used to derive meromorphic solutions to some algebraic differential equations related Painlevé equation IV, and then we illustrate our main result by some computer simulations. By the application of our result, we obtain meromorphic solutions of a nonlinear evolution equation.We can apply the idea of this study for other nonlinear evolution equations in mathematical physics.
“…where c 11 = −3 4 10 + 4a 2 10 − 2b, c 12 = c 18 = 12 4 10 z 1 − 16a 2 10 z 1 − 12 3 10 11 + 12 3 10 12 + 8a 10 11 − 8a 10 12 + 8bz 1 , c 13 = c 17 = −18 4 10 z 2 1 + 24a 2 10 z 2 1 + 48 3 10 11 z 1 − 36 3 10 12 z 1 − 32a 10 11 z 1 + 24a 10 From (7), we obtain the indeterminate relations to elliptic solutions of Equation PIV with pole at z = 0:…”
Section: Resultsmentioning
confidence: 99%
“…A great deal of research has been conducted worldwide in the study of nonlinear differential equations. [1][2][3][4][5][6][7][8][9][10] The 6 Painlevé equations have been identified as the most important nonlinear ordinary differential equations ODEs. 11 In the past 2 decades, there is continuing interest in solutions of Painlevé equations.…”
In this paper, the complex method is used to derive meromorphic solutions to some algebraic differential equations related Painlevé equation IV, and then we illustrate our main result by some computer simulations. By the application of our result, we obtain meromorphic solutions of a nonlinear evolution equation.We can apply the idea of this study for other nonlinear evolution equations in mathematical physics.
“…Consider system (17). There are nonnegative constants , , 1 , 2 , and , positive numbers , , and 1 , and functions ( ) and ( ), such that the conditions hold:…”
Section: Corollary 2 Consider System (6) With a Variable Delay ( )mentioning
confidence: 99%
“…Stability and the existence of solutions for nonlinear differential equations have been studied by many scholars [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] due to their many applications to problems in information theory, control theory, mechanics, chemistry, physics, and so on. In [5], the authors considered a second-order functional integro-differential equation with multiple delays …”
Stability of zero solution for second-order integro-differential equations with a delay is analyzed and some new results are presented. Through constructing Lyapunov functional, we give the corresponding sufficient conditions on stability of zero solution for two integro-differential equations. Moreover, an illustrative example is considered to support our new results.
“…For some recent literature on Caputo type multi-term FBVPs, we mention the papers [8,9] and the references therein. In [20], we established some new positive properties of the Green's function for the Riemann-Liouville type FBVP, in which the linear operator contains two terms:…”
In this paper, we consider a Riemann-Liouville type two-term fractional differential equation boundary value problem. Some positive properties of the Green's function are deduced by using techniques of analysis. As application, we obtain the existence and multiplicity of positive solutions for a fractional boundary value problem under conditions that the nonlinearity f (t, x) may change sign and may be singular at t = 0, 1 and x = 0, and we also obtain the uniqueness results of positive solution for a singular problem by means of the monotone iterative technique.
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