THE MEAN VALUE THEOREM AND CONVERSE THEOREM OF ONE CLASS THE FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATIONS

Applied Mathematics and Mechanics (English Edition) ›› 2001, Vol. 22 ›› Issue (6): 717-723.

THE MEAN VALUE THEOREM AND CONVERSE THEOREM OF ONE CLASS THE FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATIONS

同小军^1,2, 同登科², 陈绵云¹

1. Department of Automation Control, Huazhong University of Science and Technology, Wuhan 430074, P.R.China;
2. Department of Applied Mathematics, University of Petroleum, Dongying, Shandong 257062, P.R.China

收稿日期:1999-10-29 修回日期:2001-01-28 出版日期:2001-06-18 发布日期:2001-06-18
通讯作者: LIN Zong-chi
基金资助:
the National Natural Science Foundation of China (79979925)

THE MEAN VALUE THEOREM AND CONVERSE THEOREM OF ONE CLASS THE FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATIONS

TONG Xiao-jun^1,2, TONG Deng-ke², CHEN Mian-yun¹

1. Department of Automation Control, Huazhong University of Science and Technology, Wuhan 430074, P.R.China;
2. Department of Applied Mathematics, University of Petroleum, Dongying, Shandong 257062, P.R.China

Received:1999-10-29 Revised:2001-01-28 Online:2001-06-18 Published:2001-06-18
Supported by:
the National Natural Science Foundation of China (79979925)

摘要/Abstract

摘要： For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential equations of the second-order. So it is important to develop Asgeirsson mean value theorem. The mean value of solution for the higher order equation has been discussed primarily and has no exact result at present. The mean value theorem for the higher order equation can be deduced and satisfied generalized biaxial symmetry potential equation by using the result of Asgeirsson mean value theorem and the properties of derivation and integration. Moreover, the mean value formula can be obtained by using the regular solutions of potential equation and the special properties of Jacobi polynomials. Its converse theorem is also proved. The obtained results make it possible to discuss on continuation of the solutions and well posed problem.

Abstract: For the formal presentation about the definite problems of ultra-hyperbolic equations, the famous Asgeirsson mean value theorem has answered that Cauchy problems are ill-posed to ultra-hyperbolic partial differential equations of the second-order. So it is important to develop Asgeirsson mean value theorem. The mean value of solution for the higher order equation has been discussed primarily and has no exact result at present. The mean value theorem for the higher order equation can be deduced and satisfied generalized biaxial symmetry potential equation by using the result of Asgeirsson mean value theorem and the properties of derivation and integration. Moreover, the mean value formula can be obtained by using the regular solutions of potential equation and the special properties of Jacobi polynomials. Its converse theorem is also proved. The obtained results make it possible to discuss on continuation of the solutions and well posed problem.

中图分类号:

O175.27

同小军;同登科;陈绵云. THE MEAN VALUE THEOREM AND CONVERSE THEOREM OF ONE CLASS THE FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATIONS[J]. Applied Mathematics and Mechanics (English Edition), 2001, 22(6): 717-723.

TONG Xiao-jun;TONG Deng-ke;CHEN Mian-yun. THE MEAN VALUE THEOREM AND CONVERSE THEOREM OF ONE CLASS THE FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATIONS[J]. Applied Mathematics and Mechanics (English Edition), 2001, 22(6): 717-723.

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THE MEAN VALUE THEOREM AND CONVERSE THEOREM OF ONE CLASS THE FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATIONS

THE MEAN VALUE THEOREM AND CONVERSE THEOREM OF ONE CLASS THE FOURTH-ORDER PARTIAL DIFFERENTIAL EQUATIONS

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