Media type: E-Article Title: Monogenic functions in the biharmonic boundary value problem Contributor: Gryshchuk, Serhii V.; Plaksa, Sergiy A. imprint: Wiley, 2016 Published in: Mathematical Methods in the Applied Sciences Language: English DOI: 10.1002/mma.3741 ISSN: 0170-4214; 1099-1476 Keywords: General Engineering ; General Mathematics Origination: Footnote: Description: <jats:p>We consider a commutative algebra <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/mma3741-math-0001.png" xlink:title="urn:x-wiley:mma:media:mma3741:mma3741-math-0001" /> over the field of complex numbers with a basis {<jats:italic>e</jats:italic><jats:sub>1</jats:sub>,<jats:italic>e</jats:italic><jats:sub>2</jats:sub>} satisfying the conditions <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/mma3741-math-0002.png" xlink:title="urn:x-wiley:mma:media:mma3741:mma3741-math-0002" />, <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/mma3741-math-0003.png" xlink:title="urn:x-wiley:mma:media:mma3741:mma3741-math-0003" />. Let <jats:italic>D</jats:italic> be a bounded domain in the Cartesian plane <jats:italic>x</jats:italic><jats:italic>O</jats:italic><jats:italic>y</jats:italic> and <jats:italic>D</jats:italic><jats:sub><jats:italic>ζ</jats:italic></jats:sub>={<jats:italic>x</jats:italic><jats:italic>e</jats:italic><jats:sub>1</jats:sub>+<jats:italic>y</jats:italic><jats:italic>e</jats:italic><jats:sub>2</jats:sub>:(<jats:italic>x</jats:italic>,<jats:italic>y</jats:italic>)∈<jats:italic>D</jats:italic>}. Components of every monogenic function Φ(<jats:italic>x</jats:italic><jats:italic>e</jats:italic><jats:sub>1</jats:sub>+<jats:italic>y</jats:italic><jats:italic>e</jats:italic><jats:sub>2</jats:sub>) = <jats:italic>U</jats:italic><jats:sub>1</jats:sub>(<jats:italic>x</jats:italic>,<jats:italic>y</jats:italic>)<jats:italic>e</jats:italic><jats:sub>1</jats:sub>+<jats:italic>U</jats:italic><jats:sub>2</jats:sub>(<jats:italic>x</jats:italic>,<jats:italic>y</jats:italic>)<jats:italic>i</jats:italic><jats:italic>e</jats:italic><jats:sub>1</jats:sub>+<jats:italic>U</jats:italic><jats:sub>3</jats:sub>(<jats:italic>x</jats:italic>,<jats:italic>y</jats:italic>)<jats:italic>e</jats:italic><jats:sub>2</jats:sub>+<jats:italic>U</jats:italic><jats:sub>4</jats:sub>(<jats:italic>x</jats:italic>,<jats:italic>y</jats:italic>)<jats:italic>i</jats:italic><jats:italic>e</jats:italic><jats:sub>2</jats:sub> having the classic derivative in <jats:italic>D</jats:italic><jats:sub><jats:italic>ζ</jats:italic></jats:sub> are biharmonic functions in <jats:italic>D</jats:italic>, that is, Δ<jats:sup>2</jats:sup><jats:italic>U</jats:italic><jats:sub><jats:italic>j</jats:italic></jats:sub>(<jats:italic>x</jats:italic>,<jats:italic>y</jats:italic>) = 0 for <jats:italic>j</jats:italic> = 1,2,3,4. We consider a Schwarz‐type boundary value problem for monogenic functions in a simply connected domain <jats:italic>D</jats:italic><jats:sub><jats:italic>ζ</jats:italic></jats:sub>. This problem is associated with the following biharmonic problem: to find a biharmonic function <jats:italic>V</jats:italic>(<jats:italic>x</jats:italic>,<jats:italic>y</jats:italic>) in the domain <jats:italic>D</jats:italic> when boundary values of its partial derivatives <jats:italic>∂</jats:italic><jats:italic>V</jats:italic>/<jats:italic>∂</jats:italic><jats:italic>x</jats:italic>, <jats:italic>∂</jats:italic><jats:italic>V</jats:italic>/<jats:italic>∂</jats:italic><jats:italic>y</jats:italic> are given on the boundary <jats:italic>∂</jats:italic><jats:italic>D</jats:italic>. Using a hypercomplex analog of the Cauchy‐type integral, we reduce the mentioned Schwarz‐type boundary value problem to a system of integral equations on the real axes and establish sufficient conditions under which this system has the Fredholm property. Copyright © 2015 John Wiley & Sons, Ltd.</jats:p>