We study the integral equation related to the three and higher dimensional superradiance problem. Collective radiation phenomena has attracted the attention of many physicists and chemists since the pioneering work of R. H. Dicke in 1954. We first consider the three-dimensional superradiance problem and find a differential operator that commutes with the integral operator related to the problem. We find all the eigenfunctions of the differential operator and obtain a complete set of eigensolutions for the three-dimensional superradiance problem. Generalization of the three-dimensional superradiance integral equation is provided. A commuting differential operator is found for this generalized problem. For the three dimensional superradiance problem, an alternative set of complete eigenfunctions is also provided. The kernel for the superradiance problem when restricted to one-dimension is the same as appeared in the works of Slepian, Landau and Pollak. The uniqueness of the differential operator commuting with that kernel is indicated. Finally, a concentration problem for the signals which are bandlimited in disjoint frequency-intervals is considered. The problem is to determine which bandlimited signals lose the smallest fraction of their energy when restricted in a given time interval. A numerical algorithm for solution and convergence theorems are given. Orthogonality properties of analytically extended eigenfunctions over L2(−∞,∞) are also proved. Numerical computations are carried out in support of the theory.

In this paper, the property of Curvelet transforms for preconditioning the migration and normal operators is investigated. These operators belong to the class of Fourier integral operators and pseudo-differential operators, respectively. The effect of this preconditioner is shown in term of improvement of sparsity, convergence rate, number of iteration for the Krylov-subspace solver and clustering of singular(eigen) values. The migration operator, which we employed in this work is the common-offset Kirchoff-Born migration.

We extend our results on the degree of ill-posedness for linear integration opera- tors A with weights mapping in the Hilbert space L^2(0,1), which were published in the journal 'Inverse Problems' in 2005 ([5]). Now we can prove that the degree one also holds for a family of exponential weight functions. In this context, we empha- size that for integration operators with outer weights the use of the operator AA^* is more appropriate for the analysis of eigenvalue problems and the corresponding asymptotics of singular values than the former use of A^*A.

The paper deals with a three variable analogues of certain fractional integral operators introduced by M. Saigo. Resides giving three variable analogues of earlier known fractional integral operators of one variable as a special cases of newly defined operators, the paper establishes certain results in the form of theorems including integration by parts.

In this dissertation, a new numerical method for solving fractional calculus problems is presented. The method is based upon the fractional Taylor vector approximations. The operational matrix of the fractional integration for the fractional Taylor vector is introduced. This matrix is then utilized to reduce the solution of the fractional calculus problems to the solution of a system of algebraic equations. This method is used to solve fractional differential equations, Bagley-Torvik equations, fractional integro-differential equations, and fractional duffing problems. Illustrative examples are included to demonstrate the validity and applicability of this technique.

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We extend our results on the degree of ill-posedness for linear integration opera- tors A with weights mapping in the Hilbert space L^2(0,1), which were published in the journal 'Inverse Problems' in 2005 ([5]). Now we can prove that the degree one also holds for a family of exponential weight functions. In this context, we empha- size that for integration operators with outer weights the use of the operator AA^* is more appropriate for the analysis of eigenvalue problems and the corresponding asymptotics of singular values than the former use of A^*A.

This thesis presents a numerical approach based on generalized fractional-order Chebyshev wavelets for solving fractional-order optimal control problems. The exact value of the Riemann– Liouville fractional integral operator of the generalized fractional-order Chebyshev wavelets is computed by applying the regularized beta function. We apply the given wavelets, the exact formula, and the collocation method to transform the studied problem into a new optimization problem. The convergence analysis of the proposed method is provided. The present method is extended for solving fractional-order, distributed-order, and variable-order optimal control problems. Illustrative examples are considered to show the advantage of this method in comparison with the existing methods in the literature.