Abstract
In this paper the D’Alembert type solution for the following
Cauchy problem is constructed. Here, \(a_i\), \((i=1,2,3,4)\) and \(\phi _k\left( x\right) \), \((k=1,2,3,4)\) are given constants and functions, respectively. The cases where the roots of the characteristic equation are folded are examined and compact expressions for the solutions are obtained. The obtained solutions allow proving the uniqueness and existence of the solutions of the considered problem.
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References
Garding, L.: Linear hyperbolic partial differential equations with constant coefficients. Acta Math. 85, 1–62 (1950)
Jost, J.: Partial Differential Equations. Springer, New York (2002)
Lax, A.: On cauchy’s problem for partial differential equations with multiple characteristics. Commun. Pure Appl. Math. 9, 135–169 (1956)
Leray, J.: Hyperbolic Differential Equations. Institute for Advanced Study, Princeton (1953)
Mizohata, S.: Lectures on Cauchy Problem. Tata Institute of Fundamental Research, Bombay (1965)
Petrowski, I.G.: On cauchy problem for system of linear partial differential equations in domain of non analytic functions. Bul. Mosk. Gos. Univ. Mat. Mekh. 7, 1–74 (1938)
Pinchover, Y., Rubinstein, J.: An Introduction to Partial Differential Equations. Cambridge University Press, Cambridge (2005)
Polyanin, A.D.: Handbook of Linear Partial Differential Equations for Engineers and Scientists. Chapman and Hall/CRC Press, Boca Raton (2002)
Polyanin, A.D., Schiesser, W.E., Zhurov, A.I.: Partial differential equations. Scholarpedia 3(10), 4605 (2008)
Rasulov, M.L.: Methods of Contour Integration. North Holland, Amsterdam (1967)
Sobolev, S.L.: Some Applications of Functional Analysis in Mathematical Physics. American Mathematical Society, Providence (1963)
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The authors gratefully appreciate and acknowledge the support of International Centre for Theoretical Physics (ICTP)- Associateship Program.
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Rasulov, M., Aslan, Z., Sinsoysal, B., Bal, H. (2017). The D’Alembert Type Solution of the Cauchy Problem for the Homogeneous with Respect to Fourth Order Derivatives for Hyperbolic Equation. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2017. ICCSA 2017. Lecture Notes in Computer Science(), vol 10409. Springer, Cham. https://doi.org/10.1007/978-3-319-62407-5_54
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DOI: https://doi.org/10.1007/978-3-319-62407-5_54
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