Schwartz dağılımları vasıtasıyla lineer kısmi türevli denklemlerin çözümleri

This thesis explores Schwartz distributions, their properties, and applications to study solution of linear partial differential equations with constant coefficients. The Heaviside or Dirac delta functions are also discussed. The thesis also addresses the solution of differential equations using Fourier transforms. Furthermore, the solutions a differential equation in the space of Schwartz distributions and the iterated Laplace operator are discussed. The thesis consists of ten chapters, starting with an introduction to Schwartz distributions in the first chapter. The third chapter discusses Fourier transformations in R. The space of infinitely differentiable functions with compact support and the space of rapidly decreasing functions are examined. Next, the Fourier transform is defined on R^n. It is studied in the space L^1(R^n), the space S(R^n) is examined, and the Fourier transform on the space L^2(R^n) and on the torus is discussed. In the sixth chapter, the relationship between the Cauchy principal value integral and the Hadamard finite part is discussed. Furthermore, their application to differential equations is examined. In the seventh chapter , the Laplace operator is explored in different coordinate systems. The eighth chapter focuses on solving partial differential equations using the Fourier transform. This chapter discusses the elementary solution for the wave equation. In the last chapter, the elementary solution of iterated Laplacian is examined.