Publication: L2 - Laplace dönüşümü ile diferansiyel denklemlerin çözümleri
Abstract
L2–LAPLACE DÖNÜŞÜMÜ İLE DİFERANSİYEL DENKLEMLERİN ÇÖZÜMLERİ İntegral dönüşümleri genellikle uygulamalı matematikte, fizik ve mühendislikte kullanılmaktadır. Bu tezde integral dönüşüm teorisi ve L₂ dönüşümü ile denklem çözümü çalışılmıştır. Genelleştirilmiş Laplace dönüşümü olarak bilinen L2 Laplace dönüşümü, Laplace dönüşümü, Fourier dönüşümü, Hankel dönüşümü, Glasser dönüşümü, Stieltjes dönüşümü, Potansiyel dönüşüm, Mellin dönüşümü arasındaki bağıntılar incelenmiştir. Bu bağıntıların uygulamalı örnekleri de ele alınmıştır. L2 dönüşümü, Bessel diferansiyel denklem ve Hermite diferansiyel denklemin çözümünde yeni bir metod olarak kullanılmıştır. Bu yöntemle başka diferansiyel denklemlerin çözümlerinin de söz konusu olabileceği açıktır. Son bölümde integral dönüşümleri arasındaki bağıntılar kullanılarak bazı fonksiyonların integralleri hesaplanmıştır. Böylece bazı integral hesaplarının daha kolay elde edilebileceği gösterilmiştir. Ayrıca dönüşüm teorisi üzerine yapılan böyle bir çalışmanın devam ettirilmesiyle integral dönüşüm tablolarının genişletilebileceği açıkça görülmüştür.
SOLUTIONS OF THE DIFFERENTIAL EQUATIONS WITH L2–LAPLACE TRANSFORM The integral transforms are generally used in applied mathematics, physics and engineering. In this thesis, the integral transform theory and equation solution with L2-transform are studied. The relations between the L2-transform known as the generalized Laplace transform and the Laplace, Fourier, Henkel, Glasser, Stieltjes, Potential, Mellin transforms are examined. Same relevant examples of these relations are also taken into consideration. L2-transform is used on solution of the Bessel differential equation and the Hermite differential equation as a new method. It is obvious that solutions of the another differential equations with this method may also be in question. In the last section, integrals of some functions are computed by using the relation between these transforms. Hence, it is shown that the estimations of some integrals are obtained easily. Moreover, it is obvious that tables of integral transforms can be expanded by continuing such a study in transform theory.
SOLUTIONS OF THE DIFFERENTIAL EQUATIONS WITH L2–LAPLACE TRANSFORM The integral transforms are generally used in applied mathematics, physics and engineering. In this thesis, the integral transform theory and equation solution with L2-transform are studied. The relations between the L2-transform known as the generalized Laplace transform and the Laplace, Fourier, Henkel, Glasser, Stieltjes, Potential, Mellin transforms are examined. Same relevant examples of these relations are also taken into consideration. L2-transform is used on solution of the Bessel differential equation and the Hermite differential equation as a new method. It is obvious that solutions of the another differential equations with this method may also be in question. In the last section, integrals of some functions are computed by using the relation between these transforms. Hence, it is shown that the estimations of some integrals are obtained easily. Moreover, it is obvious that tables of integral transforms can be expanded by continuing such a study in transform theory.