Geodesic
Correct and self-adjoint problems for biquadratic operators
Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XIX, Tome 387 (2011), pp. 145-162

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In this paper we continue the theme which has been investigated in [11, 12] and [13] and we present a simple method to prove correctness and self-adjointness of the operators of the form $B^4$ corresponding to some boundary value problems. We also give representations for the unique solutions for these problems. The algorithm is easy to implement via computer algebra systems. In our examples, Derive and Mathematica were used. 
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I. N. Parasidis; P. C. Tsekrekos; T. G. Lokkas. Correct and self-adjoint problems for biquadratic operators. Zapiski Nauchnykh Seminarov POMI, Representation theory, dynamical systems, combinatorial methods. Part XIX, Tome 387 (2011), pp. 145-162. http://geodesic.mathdoc.fr/item/ZNSL_2011_387_a5/