Priyanjana M. N. Dharmawardane | Kyushu University (original) (raw)

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Papers by Priyanjana M. N. Dharmawardane

The solution of the Lane-Emden equation is very important to understand the interior stellar stru... more The solution of the Lane-Emden equation is very important to understand the interior stellar structure, and is of great importance in Mathematics since the equation of index grater than one represents one class of non-linear differential equations. The complete solution of the differential equation can be expressed as an infinite Taylor series of even powers under the boundary conditions to be imposed at the center of the star. Department of Mathematics of University of Kelaniya has found an algorithm [I] which can be used to obtain successive coefficients of the Taylor series and in this paper the proof of this algorithm will be given. The Lane – Emden equation of index m [II] has the following form. 0 2 2 2 = + + m y dx dy x dx y d (1) where m is a parameter. The boundary conditions are y(0) = 1, 0 1 ) 0 ( = = x dx dy y = 0. LaneEmden equation is unchanged when –x is substituted for x, the Taylor’s expansion contains only even powers of x and the derivatives of odd order evaluated...

Conference Publications, 2013

In this paper, we consider a quasi-linear hyperbolic systems of viscoelasticity. This system has ... more In this paper, we consider a quasi-linear hyperbolic systems of viscoelasticity. This system has dissipative properties of the memory type and the friction type. The decay property of this system is of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted energy method. Moreover, we combine this time-weighted energy method with the semigroup argument to obtain the global existence and sharp decay estimate of solutions under the smallness conditions and enough regularity assumptions on the initial data.

Nonlinear Dynamics in Partial Differential Equations

Kyoto Journal of Mathematics, 2011

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2011

ABSTRACT The aim in this paper is to develop the time-weighted energy method for quasi-linear hyp... more ABSTRACT The aim in this paper is to develop the time-weighted energy method for quasi-linear hyperbolic systems of viscoelasticity. As a consequence, we prove the global existence and decay estimate of solutions for the space dimension ngeq2n\geq 2ngeq2, provided that the initial data are small in the L2L^{2}L2-Sobolev space.

SIAM Journal on Mathematical Analysis, 2012

ABSTRACT This chapter is devoted to the study of the sharp decay estimates of solutions for quasi... more ABSTRACT This chapter is devoted to the study of the sharp decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity in the whole space. We develop the time-weighted energy method for our system, which can yield the decay estimate of solutions for small initial data in L2L^2L2, provided that ngeq2n\geq 2ngeq2. Also, we discuss the fundamental solutions to the linearized system and study the decay properties for the corresponding solution operators. Then, by employing the same time-weighted energy method together with the semigroup argument, we show the optimal decay estimate of solutions for small initial data in L2capL1L^2\cap L^1L2capL1 and for all ngeq1n\geq 1ngeq1.

Journal of Mathematical Analysis and Applications, 2010

Journal of Hyperbolic Differential Equations, 2013

We consider quasi-linear hyperbolic systems that describe the motion of viscoelastic materials. T... more We consider quasi-linear hyperbolic systems that describe the motion of viscoelastic materials. The decay property for this system exhibits a loss of regularity, and we establish a global existence theory together with sharp decay estimates, under a smallness condition and under sufficient regularity on the initial data. We combine together various tools of analysis, especially energy estimates in Fourier space, the time-weighted energy method, and the semi-group method.

The solution of the Lane-Emden equation is very important to understand the interior stellar stru... more The solution of the Lane-Emden equation is very important to understand the interior stellar structure, and is of great importance in Mathematics since the equation of index grater than one represents one class of non-linear differential equations. The complete solution of the differential equation can be expressed as an infinite Taylor series of even powers under the boundary conditions to be imposed at the center of the star. Department of Mathematics of University of Kelaniya has found an algorithm [I] which can be used to obtain successive coefficients of the Taylor series and in this paper the proof of this algorithm will be given. The Lane – Emden equation of index m [II] has the following form. 0 2 2 2 = + + m y dx dy x dx y d (1) where m is a parameter. The boundary conditions are y(0) = 1, 0 1 ) 0 ( = = x dx dy y = 0. LaneEmden equation is unchanged when –x is substituted for x, the Taylor’s expansion contains only even powers of x and the derivatives of odd order evaluated...

Conference Publications, 2013

In this paper, we consider a quasi-linear hyperbolic systems of viscoelasticity. This system has ... more In this paper, we consider a quasi-linear hyperbolic systems of viscoelasticity. This system has dissipative properties of the memory type and the friction type. The decay property of this system is of the regularity-loss type. To overcome the difficulty caused by the regularity-loss property, we employ a special time-weighted energy method. Moreover, we combine this time-weighted energy method with the semigroup argument to obtain the global existence and sharp decay estimate of solutions under the smallness conditions and enough regularity assumptions on the initial data.

Nonlinear Dynamics in Partial Differential Equations

Kyoto Journal of Mathematics, 2011

Proceedings of the Japan Academy, Series A, Mathematical Sciences, 2011

ABSTRACT The aim in this paper is to develop the time-weighted energy method for quasi-linear hyp... more ABSTRACT The aim in this paper is to develop the time-weighted energy method for quasi-linear hyperbolic systems of viscoelasticity. As a consequence, we prove the global existence and decay estimate of solutions for the space dimension ngeq2n\geq 2ngeq2, provided that the initial data are small in the L2L^{2}L2-Sobolev space.

SIAM Journal on Mathematical Analysis, 2012

ABSTRACT This chapter is devoted to the study of the sharp decay estimates of solutions for quasi... more ABSTRACT This chapter is devoted to the study of the sharp decay estimates of solutions for quasi-linear hyperbolic systems of viscoelasticity in the whole space. We develop the time-weighted energy method for our system, which can yield the decay estimate of solutions for small initial data in L2L^2L2, provided that ngeq2n\geq 2ngeq2. Also, we discuss the fundamental solutions to the linearized system and study the decay properties for the corresponding solution operators. Then, by employing the same time-weighted energy method together with the semigroup argument, we show the optimal decay estimate of solutions for small initial data in L2capL1L^2\cap L^1L2capL1 and for all ngeq1n\geq 1ngeq1.

Journal of Mathematical Analysis and Applications, 2010

Journal of Hyperbolic Differential Equations, 2013

We consider quasi-linear hyperbolic systems that describe the motion of viscoelastic materials. T... more We consider quasi-linear hyperbolic systems that describe the motion of viscoelastic materials. The decay property for this system exhibits a loss of regularity, and we establish a global existence theory together with sharp decay estimates, under a smallness condition and under sufficient regularity on the initial data. We combine together various tools of analysis, especially energy estimates in Fourier space, the time-weighted energy method, and the semi-group method.