Estimations of the Upper Bound for the Eigen-Functions of the Fourth Order Boundary Value Problem with Smooth Coefficients

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1 Math. Sci. Lett. 6, No., (7) 67 Mathematical Sciences Letters An International Journal Estimations of the Upper Bound for the Eigen-Functions of the Fourth Order Boundary Value Problem with Smooth Coefficients Karwan H. F. Jwamer,, and Rando R. Q. Rasul 3 Department of Mathematics, College of Science, University of Sulaimani, Kurdistan region, Iraq. College of Engineering, Komar University of Science and Technology, Sulaimani, Kurdistan Region, Iraq. 3 Department of Mathematics, College of basic Education, University of Sulaimani, Kurdistan region, Iraq. Received: 4 Feb. 6, Revised: 9 Sep. 6, Accepted: 5 Sep. 6 Published online: Jan. 7 Abstract: In this paper we consider a fourth-order boundary value problem with smooth coefficients. We found new expansions of linearly independent solutions that satisfy an initial condition. Then, by using these linearly independent solutions we found expressions for the eigenfunctions, and we also found an upper bound for the eigenfunctions and their derivatives. Keywords: eigenvalue problem, spectral parameter, initial condition, boundary condition, fundamental system, eigenfunction, upper bound. Introduction It is well-known that many topics in mathematical physics require the investigation of eigenvalues and eigenfunctions of Sturm Liouville type boundary value problems. In recent years, many researchers are interested in the continuous Sturm Liouville problem as we see in G. Hikmet, N. B. Kerimov, and U. Kaya in the article. Also, H. Menken in the article considered a nonself-adjoint fourth-order differential operator with periodic and anti-periodic boundary conditions. They found asymptotic formulas for the eigenvalues and eigenfunctions. Many authors investigated the method of determining a bound for eigenfunctions of a boundary value problem and its derivatives. As we see in 3,4 the authors found a bound for the eigenfunction and its derivatives of a spectral problem of the form -y + q(x)y = λ ρ(x)y. Various physics applications of this kind of a problem are found in the literature, including some boundary value problems with transmission conditions that arise in the theory of heat and mass transfer (see 5, 6). The literature on such results is voluminous, and we refer to 7. Fourth-order continuous boundary value problems with eigenfunctions dependent boundary conditions and with two supplementary transmission conditions at the point of continuity have been investigated in 8. We shall consider the fourth-order differential equation: l(y)=y (4) (x)+q(x)y(x)=λ 4 y(x),x,a () y ( j) ()= j =, U j (y)= 4 i= (iw j) i y (4 i) (a,λ)= j =,3 This paper tries to estimate a new expression for the four linearly independent solutions of the differential equation () as well as their derivatives. In section, we found an expression for the eigenfunction of the boundary value problem ()-(), which is a linear combination of the linearly independent solutions. Also, in section 3-4, we obtain upper bounds for the eigenfunction and its derivatives that we obtained in section. () Corresponding author Karwan.jwamer@univsul.edu.iq c 7 NSP

2 68 K. Jwamer, R. Rasul: Estimations of the upper bound for the eigen-functions... Fundamental System of Solutions of the Differential equation In this section, we find the expressions of four linearly independent solutions and their derivatives which satisfy the initial condition (4). These solutions mentioned in 9 for higher order. In (7, pp. 9) and ( 8, pp.5) two linearly independent solutions are found for the second order boundary value problem which is generated by the differential operator l(y) = y (x)+q(x)y(x) and they are used in a view to providing the existence of the eigenvalue for the problem l(y) = λ y(x). Here, we use the same technique to find four linearly independent solutions for the fourth-order differential equation (). Also, we use these solutions to find a bound for the eigenfunctions of ()-(). Theorem.. Consider the linear differential equation of fourth order y (4) (x)+q(x)y(x)=λ 4 y(x) (3) Where q(x) is a smooth function on,a, then (3) has the fundamental system of solutions, y (x,λ),y (x,λ), y (x,λ),y 3 (x,λ), that satisfy the initial condition Where, y (n) i (,λ)= {,i=n,i n y (x,λ) = x coshλx+cosλ x+ 3 sinλ(x ξ) +sinhλ(x ξ) q(ξ)y (x,λ)dξ. (5) y (x,λ) = sinhλx+sinλ x+ +sinhλ(x ξ) λ 3 y (x,λ) = coshλ x cosλ x+ +sinhλ(x ξ) (4) sinλ(x ξ) q(ξ)y (x,λ)dξ. (6) λ 3 y 3 (x,λ) = 3sinhλx sinλ x+ 3 +sinhλ(x ξ) sin λ(x ξ) q(ξ)y (x,λ)dξ. (7) sin λ(x ξ) q(ξ)y 3 (x,λ)dξ. (8) Proof. Consider the linear differential operator l(y)=y (4) (x)+q(x)y(x) (9) We want to find a non-zero solutions for l(y) λ 4 y= () that satisfy the initial condition (4). First we reduce () to an integro-differential equation of the form y (4) λ 4 y=m(y),m(y)= q(x)y. () The homogeneous linear differential equation y (4) λ 4 y = has for λ the solutions e λ w x,e λ w x,e λ w x,e λ w 3x. where w =,w = i,w =, w 3 = i. Then by using the method variation of parameters we can express the solutions of () for k=,,,3 as y k (x,λ)=c e λ x + c e iλ x + c e λ x + c 3 e iλ x + sinλ(x ξ)+sinhλ(x ξ) q(ξ)y 3 (x,λ)dξ. () Now, applying () in (4), gives: y (,λ)=,y (,λ)=,y (,λ)=,y (,λ)= Which gives a system of the form: c i i c c = (3) i i c 3 Solving it for c j, we obtain c j = /4 for each j = : 3, then y has the form y (x,λ) = x coshλx+cosλ x+ 3 sinλ(x ξ) +sinhλ(x ξ) q(ξ)y (x,λ)dξ. (4) Hence (5) is hold, and we can use the same technique for y (x,λ),y (x,λ),y 3 (x,λ) and we get (6), (7), (8). Lemma.. The first derivatives of the solutions of the differential equation (3) can be represented as the following expressions y (x,λ) = x λsinhλx sinλ x+ cosλ(x ξ) +coshλ(x ξ) q(ξ)y (x,λ)dξ. (5) y (x,λ) = x coshλx+cosλ x+ cosλ(x ξ) +coshλ(x ξ) q(ξ)y (x,λ)dξ. (6) y (x,λ) = sinhλx+sinλ x+ +coshλ(x ξ) cosλ(x ξ) q(ξ)y (x,λ)dξ. (7) c 7 NSP

3 Math. Sci. Lett. 6, No., (7) / 69 y 3(x,λ) = coshλ x cosλ x+ +coshλ(x ξ) cosλ(x ξ) q(ξ)y (ξ,λ)dξ. (8) Lemma.. The second derivatives of the solutions of the differential equation (3) can be represented as the following expressions y (x,λ) = λ coshλ x cosλ x+ sinλ(x ξ) +sinhλ(x ξ) q(ξ)y (ξ,λ)dξ. (9) y (x,λ) = x λsinhλx sinλ x+ sinλ(x ξ) +sinhλ(x ξ) q(ξ)y (ξ,λ)dξ. () y (x,λ) = x coshλx+cosλ x+ sinλ(x ξ) +sinhλ(x ξ) q(ξ)y (ξ,λ)dξ. () y 3 (x,λ) = sinhλx+sinλ x+ +sinhλ(x ξ) sinλ(x ξ) q(ξ)y (ξ,λ)dξ. () Lemma.3. The third derivatives of the solutions of the differential equation (3) can be represented as the following expressions (x,λ) = λ 3 sinhλ x+sinλ x+ +coshλ(x ξ) q(ξ)y (ξ,λ)dξ. (3) y (x,λ) = λ coshλ x cosλ x+ +coshλ(x ξ) q(ξ)y (ξ,λ)dξ. (4) y (x,λ) = λsinhλ x sinλ x+ +coshλ(x ξ) q(ξ)y (ξ,λ)dξ. (5) y 3 (x,λ) = coshλ x+cosλ x+ +coshλ(x ξ) q(ξ)y (ξ,λ)dξ. (6) y 3 Estimations for bounds of eigenfunctions of boundary value problem ()-() In this section, we found an upper bound for the eigenfunctions of the problem ()-(), which can be represented as a linear combination of the solutions (5)-(8). But first, we studied the simplicity of the eigenfunctions of the boundary value problem ()-(). Theorem 3. The eigenfunctions of the boundary value problem ()-() are simple. Proof. let λ be an eigenvalue for the boundary value problem ()-(). On the contrary, we suppose that φ and φ are two linearly independent eigenfunctions corresponding to λ, then φ and φ satisfy the differential equation and the boundary conditions. Now, from first and second boundary conditions we get φ (,λ)+φ (,λ)= (7) φ (,λ)+φ (,λ)= (8) which can be represented as the system: φ (,λ) φ (,λ) φ (,λ) φ = (9) Since so system (9) has non-zero solutions, and from the theory of systems we found that φ (,λ)φ (,λ) φ (,λ)φ (,λ) =. Again, from the theory of systems of differential equations, we get that φ (x,λ) and φ (x,λ) are linearly dependent at x =. Then, φ (x,λ) are linearly dependent for every x, which contradicts our assumption. Theorem 3.. If λ = r+it is an eigenvalue of boundary value problem ()-() and φ n (x,λ) is the eigen-function of the boundary value problem()-() corresponding to λ, and q(ξ) φ(ξ,λ) dξ < Then, where, max x,a φ n (x,λ) Ce q(ξ) φ(ξ,λ) dξ C= M 3 t 3 M 3 r 3, r t, r t (3) (3) Proof. If φ n (x) is an eigen-function of the differential equation, which implies that it is a solution for the differential equation, then it can be written as a linear combination of y,y,y,y 3. That is, there exists constants a,a,a,a 3 such that: φ n (x,λ)=a y (x,λ)+a y (x,λ)+a y (x,λ)+a 3 y 3 (x,λ) (3) c 7 NSP

4 7 K. Jwamer, R. Rasul: Estimations of the upper bound for the eigen-functions... where y (x,λ),y (x,λ),y (x,λ) and y 3 (x,λ) are the fundamental system of solutions in the Theorem (). Also, φ n satisfies the boundary condition (). From first boundary condition, we get φ n (,λ)=a y (,λ) +a y (,λ)+a y (,λ) +a 3 y 3 (,λ)=. and from second boundary condition, we get φ n (,λ)=a y (,λ) +a y (,λ)+a y (,λ) +a 3 y 3(,λ)=. Hence a = and a =. Thus, the eigenfunction and its derivatives reduce to the following expressions φ n (x,λ) = a coshλ x cosλ x+a 3 3sinhλ x sinλ x+ 3 sin λ(x ξ)+sinhλ(x ξ) q(ξ)φ n (ξ,λ)dξ (33) φ n(x,λ) = a sinhλx+sinλ x+a 3 coshλ x cosλ x+ cosλ(x ξ)+coshλ(x ξ) q(ξ)φ n (ξ,λ)dξ (34) n (x,λ) = a coshλx+cosλ x+a 3 sinhλx+ sinλx+ sinλ(x ξ)+sinhλ(x ξ) q(ξ)φ n (ξ,λ)dξ (35) φ n (x,λ) = a λsinhλ x sinλ x +a 3 coshλ x+cosλ x + φ +coshλ(x ξ) q(ξ)φ n (ξ,λ)dξ (36) Also, from second boundary condition, we obtain: U (φ n (x)) = a λ sinλa+iλ cosλ a +a 3 cosλ a+i sinλ a + isinλ(a ξ) cosλ(a ξ) q(ξ)φ n (ξ,λ)dξ = (37) Now, since e x is continuous in,a, so it has a maximum value in,a. We put max ξ,a e ξ = M. So e ξ M, and since t, r then e t ξ M t and e r ξ M r. And since t, r R, so we have t r or r t, if r t then ξ r ξ t for all ξ,a, then e ξ r e ξ t M t. If I =,then I = So And,then isinλ(a ξ) cosλ(a ξ)q(ξ)φ n (ξ,λ)dξ (38) isinλ(a ξ) cosλ(a ξ)q(ξ)φ n (ξ,λ)dξ sinλ(a ξ) + cosλ(a ξ) q(ξ) φ n (ξ,λ) dξ e t (a ξ) + e t (a ξ) q(ξ) φ n (ξ,λ) dξ M t q(ξ) φ n (ξ,λ) dξ. I M t q(ξ) φ n (ξ,λ) dξ. (39) G = isinλ a+cosλ a So = i eiλ a e iλ a i G=i sinλa+cosλ a (4) + eiλ a + e iλ a = eiλ a + e iλ a + e iλ a + e iλ a = e iλ a + e iλ a = e iλ a = e t ir a = e ta e t a M t. G M t. (4) Then, from (37) we obtain an expression for a in terms of a 3 as follows a λ sinλa +iλ cosλ a +a 3 cosλ a+i sinλ a I a = I a 3cosλ a+i sinλ a iλi sinλa+cosλ a = a 3G I iλ G. =. c 7 NSP

5 Math. Sci. Lett. 6, No., (7) / 7 Thus, a = a 3G I iλ G. (4) So, the eigenfunction of the boundary value problem ()- () has the following form φ n (x,λ) = G I iλ G sinλx+ 3 q(ξ)φn (ξ,λ)dξ coshλx cosλ x+ 3sinhλ x sinλ(x ξ)+sinhλ(x ξ). (43) Now, we want to find the maximum of φ n (x,λ). We have φ(x,λ) = i( + I G ) 3coshλ x cosλ x That means, + 3sinhλx sinλ x + 3 sinλ(x ξ) +sinhλ(x ξ)q(ξ)φ n (ξ,λ)dξ. φ(x,λ) 3 i( + I G ) 3 coshλ x + cosλ x + 3 sinhλx + sinλ x + 3 sinλ(x ξ) + sinhλ(x ξ) q(ξ) φ n (ξ,λ) dξ i( + I ) coshλx + cosλ x G + sinhλ x + sinλ x + sinλ(x ξ) + sinhλ(x ξ) q(ξ) φ n (ξ,λ) dξ If λ = r + it, and by using the following relations cosz e Imz, sinz e Imz, (44) coshz e Rez, sinhz e Rez. (45) Then, by replacing z = λ x in the above relations we get: cosλ x e t x, sinλx e t x, coshλ x e r x, sinhλx e r x. (46) and cosλ(x ξ) e t (x ξ), coshλ(x ξ) e r (x ξ), sinλ(x ξ) e t (x ξ), sinhλ(x ξ) e r (x ξ).(47) Therefore, (44) has the following form: φ n (x,λ) M t I 3+ G 3 Again, from (39) and (4), we get : q(ξ) φ n (ξ,λ) dξ. (48) φ n (x,λ) Ce 3C q(ξ) φ n (ξ,λ) dξ. (49) Where, C = M3 t 3. And since we choose x as any arbitrary number in, a, then max x,a φ n (x,λ) Ce 3C q(ξ) φ n (ξ,λ) dξ. (5) 4 Estimations for bounds of the derivatives of the eigenfunctions of boundary value problem ()-() This section studies the method of finding a bound for the derivatives of eigenfunctions for the problem ()-(). Theorem 3.. If λ = r+it is an eigenvalue of boundary value problem ()-() and φ n (x,λ) is the eigen-function of the boundary value problem()-() corresponding to λ, and q(ξ) φ(ξ,λ) dξ < Then, max x,a φ ( j) n M 3 t C, r t (x,λ) 3 j M 3 r C, r t. (5) 3 j for j =,,3. where, C= q(ξ) φ(ξ,λ) dξ. Proof. Since from (43) we have φ n (x,λ) = G I iλ G sinλx+ 3 q(ξ)φn (ξ,λ)dξ coshλ x cosλ x+ 3sinhλx sinλ(x ξ)+sinhλ(x ξ). (5) Then we can find the first, second and third derivatives of φ n (x,λ) in,a, which they have the following expressions: φ n(x,λ) = G I sinhλx+sinλ x+ iλ G coshλ x cosλ x+ cosλ(x ξ) +coshλ(x ξ) q(ξ)φ n (ξ,λ)dξ. (53) c 7 NSP

6 7 K. Jwamer, R. Rasul: Estimations of the upper bound for the eigen-functions... φ n (x,λ) = G I coshλ x+cosλ x+ iλ G sinhλx +sinλ x+ sinλ(x ξ) +sinhλ(x ξ) q(ξ)φ n (ξ,λ)dξ. (54) By using (39) and (4), the right-hand side of the above inequality reduces to M t + M3 t q(ξ) φ n (ξ,λ) dξ q(ξ) φ n (ξ,λ) dξ. (58) φ n (x,λ) = G I ig +cosλ x+ sinhλx sinλ x+ coshλx +coshλ(x ξ) q(ξ)φ n (ξ,λ)dξ. (55) Finally, we obtain φ M t n (x,λ) + M3 t q(ξ) φ n (ξ,λ) dξ q(ξ) φ n (ξ,λ) dξ. (59) φ n (x,λ) = G I sinhλx+sinλ x+ iλ G coshλx cosλ x + cosλ(x ξ)+coshλ(x ξ) q(ξ)φn (ξ,λ)dξ. G I iλ G sinhλx + sinλ x + coshλx cosλ x + cosλ(x ξ) + coshλ(x ξ) q(ξ) φn (ξ,λ) dξ. + I G M t + M t q(ξ) φ n (ξ,λ) dξ. (56) Simple calculation of the above inequality gives the following inequality φ n(x,λ) M t + I G q(ξ) φ n (ξ,λ) dξ. (57) By calculating the right-hand side of this inequality as follows φ M t n (x,λ) +( M3 t ) q(ξ) φ n (ξ,λ) dξ. M3 t +(3 M3 t M3 t ) q(ξ) φ n (ξ,λ) dξ. +3 q(ξ) φ n (ξ,λ) dξ M3 t C where, C=+3 q(ξ) φ n(ξ,λ) dξ. This means that: φ M3 t n (x,λ) C Since x is arbitrary in,a then we can say that max x,a φ n(x,λ) M3 t C. (6) c 7 NSP

7 Math. Sci. Lett. 6, No., (7) / 73 From (53), we have φ n (x,λ) = G I coshλ x+cosλ x+ iλ G sinhλx +sinλ x+ sinλ(x ξ) +sinhλ(x ξ) q(ξ)φ n (ξ,λ)dξ + I G coshλx + cosλ x + sinhλx + sinλ x + sin λ(x ξ) + sinhλ(x ξ) q(ξ) φn (ξ,λ) dξ (6) Simple calculation leads to φ n(x,λ) M t + I M t + G M t + M3 t q(ξ) φ n (ξ,λ) dξ. q(ξ) φ n (ξ,λ) dξ q(ξ) φ n (ξ,λ) dξ By using the same techniques as we did for φ n(x,λ), we obtain φ n(x,λ) M t +( M3 t M3 t M3 t ) q(ξ) φ n (ξ,λ) dξ +(3 M3 t +3 ) q(ξ) φ n (ξ,λ) dξ q(ξ) φ n (ξ,λ) dξ M3 t C. Where C = +3 q(ξ) φ n(ξ,λ) dξ. Thus, we have n(x,λ) M3 t C. (6) φ Again, x is arbitrary in,a so we obtain max x,a φ n(x,λ) M3 t C. (63) Next, from (55) we have G I n (x,λ) = ig φ sinhλx sinλ x+ coshλ x +cosλ x+ +coshλ(x ξ) q(ξ)φ n (ξ,λ)dξ. G I ig sinhλx + sinλ x + coshλx + cosλ x + cosλ(x ξ) + coshλ(x ξ) q(ξ) φ n (ξ,λ) dξ. (64) Some simple calculations give: φ n (x,λ) M t + I G +M t q(ξ) φ n (ξ,λ) dξ M t + M 3 t q(ξ) φ n (ξ,λ) dξ +M t q(ξ) φ n (ξ,λ) dξ. By using the same techniques as we did for φ n(x,λ), we obtain: φ n (x,λ) M t +(M 3 t + M t ) q(ξ) φ n (ξ,λ) dξ M 3 t +(3M 3 t ) M 3 t +3 M 3 t C, q(ξ) φ n (ξ,λ) dξ q(ξ) φ n (ξ,λ) dξ where C=+3 q(ξ) φ n(ξ,λ) dξ. That is, φ n (x,λ) M 3 t C, Again, x is arbitrary in,a so we obtain max x,a φ n (x,λ) M3 t C. (65) So by the above calculations, we proved that the derivatives of the eigenfunctions of the boundary value problem ()-() are bounded. For complex number λ = r+it and r t max x,a φ ( j) n (x,λ) M3 t C, j =,,3. (66) 3 j And for the complex number λ = r+ it and r t we repeat the same process as above, and we get max x,a φ ( j) n Hence, Theorem 4 was proved. (x,λ) M3 r C, j =,,3. (67) 3 j c 7 NSP

8 74 K. Jwamer, R. Rasul: Estimations of the upper bound for the eigen-functions... 5 Conclusion In this work, we found four linearly independent solutions of the boundary value problem ()-() and then we used these solutions to get the expressions of the eigenfunctions and their derivatives. We also proved that all the eigenfunctions of the boundary value problem ()-() are simple, and finally, we obtained upper bounds for the eigenfunctions and their derivatives. References G. Hikmet and N. B.Kerimov and U. Kaya, Spectral properties of fourth order differential operators with periodic and antiperiodic boundary conditions, vol. 68, Results Math.,68, 5-58, (5). H. Menken, Accurate asymptotic formulas for eigenvalues and eigenfunctions of a boundary-value problem of fourth order, Bound. Value Probl., (). 3 K. H. Jwamer and Aryan Mohammed, On the Boundedness of the First and Second Derivatives of a Type of the Spectral Problem, vol. 4, Natural Sciences Publishing Corp, 3, 83-89, (5). 4 K. H. Jwamer and Rostam Saeed K and Khelan H Q, Spectral Properties of Second Order Differential Equations with Spectral Parameter in the Boundary Conditions, 3, 65-69, (). 5 C. Fulton, Two-point boundary value problems with eigenvalue parameter contained in the boundary conditions., 77, Proc Roy Soc Edinburgh Sec A,93-38, (977). 6 Y. Y. Titeux I, Completeness of root functions for thermal conduction in a strip with piecewise continuous coefficients, 7, Math Models Methods Appl, 35-5,(997). 7 M. S. P. Eastham, Theory of ordinary differential equations, Van Nostrand Reinhold, (97). 8 I. S. Sargsjan and B. M. Levitan, Introduction to spectral theory : selfadjoint ordinary differential operators, Rhode Island: American Mathematical Society, (975). 9 M. A. Naimark, Linear Differential Operators Part Elementary Theory of Linear Differential Operators, New York: Frederick Ungar Publishing, (967). Karwan H. F. Jwamer Professor of Mathematics, Department of College of Science at the Sulaimani University, Kurdistan Region, Sulaimani,Iraq. He obtained Ph.D from Dagestan State University, South of Russian in. His researches interests include spectral analysis for different type boundary value problems, approximation by spline functions. He supervised Seven students two Ph.D and five M.Sc. in the field of differential equations and numerical analysis. He has published over fifty eight papers in these areas and also five books. He is referee and editorial board for more than eighteen international mathematical journals. Rando Rasul is an M.S.C. student at the Mathematics Department of College of Science at the Sulaimani University, Kurdistan Region, Sulaimani, Iraq. He obtained B.S. at the Mathematics Department of Faculty of Science and Science Education School of Science at the Sulaimani University, Kurdistan Region in 3. c 7 NSP