Theory and Applications of Convolution Integral Equations
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For users of convolution integral equations, the volume contains numerous, well-classified inversion tables which correspond to the various convolutions and intervals of integration. It also has an extensive, up-to-date bibliography. The convolution integral equations which are considered arise naturally from a large variety of physical situations and it is felt that the types of solutions discussed will be usefull in many diverse disciplines of applied mathematics and mathematical physical.
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For researchers and graduate students in the mathematical and physical sciences whose work involves the solution of integral equations. Litvinchuk [ 8 ] studied a class of Wiener-Hopf type integral equations with convolution and Cauchy kernel and proved the solvability of the equation. Nakazi-Yamamoto [ 10 ] proposed a class of convolution SIEs with discontinuous coefficients and transformed the equations into a Riemann boundary value problem RBVP by Fourier transform, and given the general solutions of the equation.
Later on, Li [ 11 ] discussed the SIEs with convolution kernels and periodicity, which can be transformed into a discrete jump problems by discrete Fourier transformation, and the solvable conditions and the explicit expressions of general solutions were obtained. The purpose of this article is to extend the theory to some classes of singular integral equations of convolution type with Cauchy kernels in the class of exponentially increasing functions. Such equations can be transformed to RBVPs with either an unknown function on a straight line or two unknown functions on two parallel straight lines by Fourier transformation.
We prove the existence of the solution for the equations; moreover, the general solutions and the conditions of solvability are obtained under some conditions. Therefore, the result in this paper further generalizes the results of [ 7 — 11 ]. Lemmas 2. We introduce the following two lemmas see [ 4 ].
In this section we consider the following several classes of SIEs in the class of exponentially increasing functions, and we shall transform these equations into the generalized RBVPs. In this paper we extend the results of [ 2 , 7 ] to the class of exponentially increasing functions.
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Without loss of generality, we mainly study the SIEs of dual type. The method mentioned in this paper may also be applied to solving the other classes of equations.
Extending t in 3. As a whole, after taking the Fourier transform for 3. Case 1: Equation 3.
Case 2: Equation 3. Case 3: Equation 3. But in Case 3 it is possible that a solution of 3. By using the method of analytic continuation, this case can be transformed into Case 2, therefore, in this paper we only discuss Case 1 and Case 2.sacacolkibb.tk
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For Case 1, without loss of generality, we only study equation 3. Taking the Fourier transform on the above obtained equations, respectively, we obtain. For Case 2, we only solve equation 3. In this case, in order to solve 3. Hence, in 3. By Lemmas 2. Hence, we multiply each term of 3. Then by taking the Fourier transform on the above obtained equations, we get. Note that BVP 3.
Theory and Applications of Convolution Integral Equations
In this case, 3. By applying the Plemelj equation see [ 10 ] to 4. Putting 4. In this subsection, we shall solve 3. Hence, for the functions appearing in 3. In order to solve 3. Then the first equality of 4. Similarly, the second equality of 4. Thus, we need only to discuss 4. Thus we may denote. Therefore, by extended Liouville theory [ 13 ], we obtain. Therefore, one has.
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From the above discussions, we can obtain the solutions of 4. Therefore, a solution of 3. Next, we come to discuss the solvability conditions for equation 3. Thus, we have the following conclusion as regards the solution of equation 3. Finally, we remark that the method of this paper may be applied to solving the equations mentioned above in the non-normal cases or, the exceptional cases , that is,. As for the method of solution for this case, there is no essential difference for the solving method with the normal case.
We will not elaborate on that here. In this article, some classes of SIEs of convolution type with Cauchy kernels are solved in the class of exponentially increasing functions. By Fourier transform, such equations are transformed into RBVPs on either a straight line or two parallel straight lines. The exact solutions of equation 3. Here, our method is different from the ones for the classical boundary value problem, and it is novel and effective. Thus, the result in this paper generalizes the theory of classical boundary value problems and singular integral equations.