Graduate Thesis Or Dissertation
 

Solution by the method of G. C. Evans of the Volterra integral equation corresponding to the initial value problem for a non-homogeneous linear differential equation with constant coefficients

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  • In the first chapter of this thesis, several methods are used to solve an n-th order linear ordinary differential equation with constant coefficients together with n known initial values. The first method is the standard elementary method where the general solution of the differential system is found as a sum of two solutions u and v where u is the solution of the homogeneous part of the ordinary differential equation and v is any particular solution of the nonhomogeneous differential equation. The method is not strong enough to find a particular solution for every function that might be given as the non-homogeneous term of the ordinary differential equation and so we try a more powerful approach for finding v; hence the Lagrange's method of variation of parameters. Following this, the method of Laplace transforms is employed to solve the differential system. In the second chapter the n-th order linear ordinary differential equation is converted into a Volterra integral equation of second kind and in the next chapter, the idea of the resolvent kernel of an integral equation is introduced with some proofs of the existence and convergence of the resolvent kernel of the integral equation. The method of solving the Volterra integral equation by iteration is briefly discussed. The fourth chapter is devoted to solving the Volterra integral equation with convolution type kernel by the method of E. T. Whittaker, but the method is found to be very involved, and as a result, a method suggested by G. C. Evans (1911) is employed in calculating the resolvent kernels for kernels made up of sums of two exponential functions (the method of iteration was applied to the same problem but it was tedious--it took about 20 pages of writing) and finally the method provides an easier way for calculating the resolvent kernel of the Volterra integral equation corresponding to an n-th order linear ordinary differential equation with constant coefficients.
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