### EKSISTENSI DAN KETUNGGALAN SOLUSI PERSAMAAN GELOMBANG AIRY MENGGUNAKAN PENDEKATAN SEMIGRUP C_0

M Kiftiah, W B Partiwi, F Fran, B Prihandono

#### Abstract

Semigrup merupakan salah satu metode yang digunakan untuk menunjukkan Masalah Nilai Awal (MNA) dari persamaan diferensial di Ruang Hilbert bersifat well posed. MNA dalam abstrak ini disebut Masalah Cauchy Abstrak. Semigrup pada Ruang Hilbert H merupakan keluarga operator linear pada Ruang Hilbert H yang tertutup terhadap komposisi dan memiliki elemen identitas. Lebih lanjut, jika semigrup mempunyai turunan kanan di maka turunannya disebut infinitesimal generator. Dalam hal ini, Teorema Lumer Philips memberikan ekivalensi antara infinitesimal generator dengan semigrup. Dalam hal tertentu semigrup dapat diperluas menjadi grup. Teorema Stone memberikan ekuivalensi antara generator dengan grup. Secara teknis, Teorema Lumer Philips mengatakan MNA bersifat well posed jika dan hanya jika infinitesimal generatornya bersifat m-dissipative. Selanjutnya, pendekatan semigrup diaplikasikan pada persamaan Airy.

Semigroup is one method used to show the Initial Value Problems (MNA)of differential equations in Hilbert space is well posed. In this research, MNA called Abstract Cauchy Problems. Semigroup is a family of linear operators on Hilbert Space H which is closed under composition and has an identity element. Furthermore, if semigrup has a right derivative at then it is called infinitesimal generator. In this case, the Lumer Philips Theorem provides the equivalence between infinitesimal generator and semigroup. In certain cases, semigroup can be expanded into a group. Stone Theorem gives the equivalence between generator and group. Then Lumer Philips Theorem said the MNA is well posed if and only if the infinitesimal generator is m-dissipative. Furthermore, semigroup approach was applied to the Airy equation.

#### Keywords

generator; semigroup; Airy equation

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