PROPERTIES OF SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS FOR LAVRENT EV-BITSADZE EQUATIONS AT CORNER POINTS
|
|
- Emerald Heath
- 6 years ago
- Views:
Transcription
1 Electronic Journal of Differential Equations, Vol , No. 93, pp. 9. ISSN: URL: or ftp ejde.math.txstate.edu PROPERTIES OF SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS FOR LAVRENT EV-BITSADZE EQUATIONS AT CORNER POINTS MAKHMUD A. SADYBEKOV, NURGISSA A. YESSIRKEGENOV Abstract. We consider the Neumann-Tricomi problem for the Lavrent ev- Bitsadze equation for the case in which the elliptic part of the boundary is part of a circle. For the homogeneous equation, we introduce a new class of solutions that are not continuous at the corner points of the domain and construct nontrivial solutions in this class in closed form. For the nonhomogeneous equation, we introduce the notion of an n-regular solution and prove a criterion for the existence of such a solution.. Introduction Let R 2 be a finite domain bounded for y < by the characteristics AC : x + y = and BC : x y = of the Lavrent ev-bitsadze equation sgnyu xx + u yy = fx, y. and for y > by the circular arc σ δ = {x, y : x /2 2 +y+δ 2 = /4+δ 2, y > }. Neumann-Tricomi problem problem N-T. Find a solution of. with the boundary condition u AC =,.2 n σδ =,.3 where n = x /2 x + y + δ y. We assume that the classical transmission conditions ux, + = ux,, u y x, + = u y x,, < x <,.4 hold for the solution on the line y =, of type change of the equation. Along with problem N T, consider the adjoint problem. 2 Mathematics Subject Classification. 35M. Key words and phrases. Neumann-Tricomi problem; n-regular solution; Lavrent ev-bitsadze equation. c 24 Texas State University - San Marcos. Submitted July 9, 24. Published September 6, 24.
2 2 M. A. SADYBEKOV, N. A. YESSIRKEGENOV EJDE-24/93 Problem N-T*. Find a solution of the equation with the boundary condition sgnyυ xx + υ yy = gx, y.5 υ BC =,.6 υ n σ δ =..7 Here we also assume that the transmission conditions υx, + = υx,, υ y x, + = υ y x,, < x <,.8 are satisfied. Bitsadze [, p ] proved the existence and uniqueness of regular solution of Neumann-Tricomi problem. The completeness of eigenfunctions of the Neumann- Tricomi problem for a degenerate equation of mixed type in the elliptic part of the domain was investigated by Moiseev and Mogimi [7]. Also, they showed that a system of functions consisting of sums of Legendre functions is complete. The existence and uniqueness of a strong solution of the Tricomi problem where instead of.3, it was given by condition u σ = for the Lavrent ev-bitsadze equation were studied in [3, 4, 5]. In [8] the spectral methods of solving boundary value problems for mixed-type differential equations of second order in a 3D domain were studied. Existence and uniqueness of a solution of the Lavrent ev Bitsadze problem was proved. In [] we proved a criterion for the strong solvability of the Neumann-Tricomi problem in L 2. It was shown that if the elliptic part of the domain coincides with the semi-circle, then the Neumann-Tricomi problem in the classical domain is not strongly solvable in L 2. In [] for the Tricomi problem it was studied properties of solutions at corner points. Also, it was given a criterion for the existence of n-regular solution. Relation between the uniqueness of solution of the problem and the order of smoothness or features of solutions is well-known and it is a particularly evident in problems for degenerate equations [9] and mixed-type equations []. In this paper, we introduce new classes of solutions of the Neumann-Tricomi problem depending on the behavior at the corner points and study their properties. 2. Main results We say that a function hx, y belongs to the class C α,β A,B if and only if x α x β hx, y C. As usual, = {y > } and 2 = {y < }. A solution of the problem is understood as a function in the class C 2 C 2 2 C C σ δ. We denote the angle at which the curve σ δ approaches the line of change of type satisfies γ δ = arccot2δ, < γ δ < π. 2. Theorem 2.. There are infinitely many solutions u k x, y C α k,α k A,B C σ δ to the homogeneous problem N-T f. These solutions are given by the relations u k x, y = { Re x+iy x 2 +y α k + Im x+iy 2 x 2 +y α k for y >, 2 x y α k 2.2 for y < ;
3 EJDE-24/93 SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS 3 where α k = π + 4k/4γ δ, k =,, In addition, u k x, y / L 2 for k, and u x, y L 2 only if γ δ > π/ Theorem 2.2. There are infinitely many solutions υ k x, y C α k, α k A,B C σ δ to the homogeneous problem N-T* g, where α k is given by relation 2.3. These solutions are given by the formulas υ k x, y = { Re x+iy α k + Im x+iy x 2 +y α k for y >, 2 x y α k 2.5 for y < ; x 2 +y 2 in addition, υ k x, y / L 2 for k, and υ x, y L 2 only under condition 2.4. Proof of Theorem 2... We denote x + iy wx, y = x 2 + y 2 αk, then ux, y = Re wx, y + Im wx, y, y >. For y > and f = equation. can be written as u xx + u yy =. By a direct calculation, we have x + iy αk 2 w xx = α k α k x 2 + y 2 x4 + 4 x 3 yi 6 x 2 y 2 4 xy 3 i + y 4 x 2 + y α k x + iy x 2 + y 2 αk 2 x x 2 yi 6 xy 2 2y 3 i x 2 + y 2 3, x + iy αk 2 w yy = α k α k x 2 + y 2 x4 4 x 3 yi + 6 x 2 y xy 3 i y 4 x 2 + y α k x + iy x 2 + y 2 α 2 x 3 6 x 2 yi + 6 xy 2 + 2y 3 i x 2 + y 2 3. Thus, w xx + w yy =, since x 2 + y 2 in, hence, Rew xx + w yy + Imw xx + w yy = u xx + u yy =. For y < and f = equation. can be written as u xx u yy =. By a direct calculation, from 2.2, for y < we have αk 2 u xx x, y = α k α k x y x y 4 αk + α k x y 2 x y 3,
4 4 M. A. SADYBEKOV, N. A. YESSIRKEGENOV EJDE-24/93 αk 2 u yy x, y = α k α k x y x y 4 αk + α k x y 2 x y 3. Thus, u xx u yy =, since x + y in 2. The function in 2.2 satisfies equation. for both y > and y <. 2. By 2.2, for y <, ux, y = x y αk = x + y x y αk, we have x + y αk x+y= u AC = u x+y= = =, x y since α k >. Thus, function in 2.2 satisfies the boundary condition.2 in the hyperbolic part of the domain. The contour σ δ has the form 2yδ = x x 2 y 2. Thus, boundary condition.3 can be written as n x x 2 y 2 =2yδ =. By definition 2. for the number γ δ, we have n α k y α k σ δ = 2sin γ δ α k x2 + y 2 α cosα k γ δ sinα k γ δ. k By the definition of α k in 2.3, we obtain α k = π 4 + πk γ δ cosα k γ δ sinα k γ δ =. Thus, n σδ =. The function in 2.2 satisfies the boundary condition To check conditions.4, from the representation of 2.2, we obtain ux, = ux, + = x αk, u y x, = u y x, + = α k x αk x 2, and conditions.4 are satisfied. As a result, function in 2.2 is solution of the homogeneous Problem N-T. It is easy to see that function in 2.2 belongs to the class C α k,α k A,B C σ δ. Next, we prove the final statement of theorem 2.. u k 2 L = 2 u k x, y 2 dx dy <. 2.6 Note that u k x, y 2 dx dy = u k x, y 2 dx dy + 2 u k x, y 2 dx dy,
5 EJDE-24/93 SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS 5 u k x, y 2 x + y 2αk dx dy = dx dy 2 2 x + y 2αk < x + y dx dy. 2 Using the change of variables x + y = ξ, x y = η, we obtain u k x, y 2 dx dy < ξ 2α k dξdη = η dη ξ 2α k dξ = η 2α k dη < 22α k for 2α k >, α k <. By using the change of variables x = r2 +r cos ϕ r sin ϕ +2r cos ϕ+r, y = 2 +2r cos ϕ+r in 2, we obtain u k x, y 2 r 2αk + sin 2α k ϕr dx dy = + 2r cos ϕ + r 2 dr dϕ 2 = γδ + sin 2α k ϕdϕ r 2α k+ dr + 2r cos ϕ + r 2 2 < for 2α k < α k <. Thus, ratio 2.6 is satisfied for α k <. Taking into account the definition of α k in 2.3, it is easy to see that ratio 2.6 is satisfied only for k =, and π 4γ δ < γ δ > π 4. Theorem 2.2 can be proved in a similar way; so we omit its proof. Let us proceed to the analysis of the nonhomogeneous problem N-T and N-T*. An n-regular solution of Problem N-T N-T* is defined as a solution, ux, y C 2 C 2 2 C C σ δ C n, A,B υx, y C 2 C 2 2 C C σ δ C, n A,B. The following theorems hold for the nonhomogeneous Problems N-T and N-T*. Theorem 2.3. The solution of Problem N-T is n-regular for any right-hand side fx, y C if and only if γ δ < π/4n, n =, 2, The solution is n-regular for arbitrary approach angles γ δ only if the right-hand side of. satisfies the conditions υ k x, yfx, y dx dy =, k =,..., j, 2.8 where υ k are the functions given by 2.5, j = [nγ δ /π /4], and [z] is the integer part of z. In this case, the number of conditions 2.8 depends on the angle γ δ, and their maximum number is equal to n as γ δ π.
6 6 M. A. SADYBEKOV, N. A. YESSIRKEGENOV EJDE-24/93 Theorem 2.4. Condition 2.7 is necessary and sufficient for the n-regularity of the solution of Problem N-T* for any right-hand side gx, y C; for arbitrary approach angles γ δ, the right-hand side of.5 satisfies the relations u k x, ygx, y dx dy =, k =,..., j, 2.9 where the u k are the functions given by 2.2 and j = [nγ δ /π /4]. In this case, the number of conditions 2.9 depends on the angle γ δ, and their maximum number is equal to n as γ δ π. Remark 2.5. Conditions 2.8 and 2.9 with k are not orthogonality conditions in L 2, and for k they are orthogonality conditions only if inequality 2.4 holds. This immediately follows from theorems 2. and 2.2. Proof of Theorem 2.3. Set ux, y = τx and u y x, = νx. In the hyperbolic part of the domain 2, we consider the Cauchy-Goursat problem u xx + u yy = fx, y, u AC =, u y x, = νx. The solution of this problem has the form [2, p.2]: x+y x+y ux, y = νtdt x y dξ f ξ + η, ξ η dη. 2 ξ 2 2 Then we obtain the main relation x τx = νtdt x x dξ f ξ + η, ξ η dη, < x <. 2 ξ 2 2 It is convenient to represent it in the form x τ = x + x νx θ/ + x θ + x θ 2 dθ + F x + x, < x <, 2. where F x = x x dξ f ξ + η, ξ η dη. 2 ξ 2 2 We apply the Mellin transform F s = x s fxdx to both sides of relation 2.. By using the formula [2, p. 269] x s dx x for ux = νx/+x +x 2 and uxθυθdθ = υx = from 2., we obtain the relation x s uxdx { for x <, for x >, x s υxdx, τs = s νs + F s. 2. Here τs = x s τ x dx, νs = x s νx/ + x x + x + x 2 dx, 2.2 F s = x s x F dx x
7 EJDE-24/93 SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS 7 In the elliptic part, we consider the problem u xx + u yy = fx, y, By making the change of variables x = n σδ =, ux, = τx. r2 + r cos ϕ + 2r cos ϕ + r 2, y = r sin ϕ + 2r cos ϕ + r 2, 2.4 by using the Mellin transform, and by solving the resulting problem, we obtain where fs, ϕ = νs = tansγ δ sτs γδ u 2 s, tfs, tdt, 2.5 r s r 2 + 2r cos ϕ + r 2 2 f r 2 + r cos ϕ + 2r cos ϕ + r 2, r sin ϕ + 2r cosϕ + r 2 dr, 2.6 u 2 s, ϕ = cos sϕ + sin sϕ sin sγ δ cos sγ δ, 2.7 and the functions τs and νs are defined in 2.2. Now from relations 2. and 2.5, we obtain [ γδ ] νs = tansγ δ sf s u 2 s, tfs, tdt [ + tansγ δ ], 2.8 τs = [ sf s + γδ ] u 2 s, tfs, tdt [s + tansγ δ ]. 2.9 First, let us analyze definitions 2.2 of the functions τs and νs and their relationship with the original functions τx and νx. By making the obvious change of variables t = x/ + x, we reduce relation 2.2 to the form τs = t s t s τtdt, νs = t s t s νtdt. Hence, it follows that if the function τs is continuous on the interval,, then the function τt is continuous at the point t = and has a zero of order there. As a result, by taking into account the definitions of the functions τx and νx, for the n-regularity of the solution, the right-hand sides in relations 2.8 and 2.9 should be continuous for n < s <, whence we obtain γ δ < π/4n. Consequently, condition 2.7 is necessary and sufficient for the n-regularity of the solution of the Neumann-Tricomi problem for any right-hand side fx, y C. The proof of first part of Theorem 2.3 is completed. Now let us proceed to the proof of properties of solutions for arbitrary approach angles γ δ. Suppose that condition 2.7 fails. It follows that, for s = α k = π + 4k/4γ δ n,, for k =,..., j, and for j, which is defined in the statement of the theorem, the denominator in relations 2.8 and 2.9 is zero. Therefore, for the n-regularity, it is necessary and sufficient that the numerator is zero at these points as well, γδ sf s + u 2 s, tfs, tdt =. 2.2 s= αk
8 8 M. A. SADYBEKOV, N. A. YESSIRKEGENOV EJDE-24/93 In this equation, we take into account relation 2.3 and the following property of the Mellin transform [6, p.567]: if then gs = sgs = x s fx dx, x s xf x dx. Now, by setting s = α k, by returning to the variables x and y according to formulas 2.4, and by taking into account relations 2.3, 2.5, 2.6, and 2.7, we find that condition 2.2 can be represented in the form υ k x, yfx, ydxdy + υ k x, yfx, ydxdy + = υ k x, yfx, ydxdy =. Here k =,..., j ; moreover, the number of such k by the definition of j cannot exceed n. Theorem 2.4 can be proved in a similar way, we omit its proof. Conclusion. In this article, it has been shown that the number of solutions of the homogeneous Neumann-Tricomi problem admitting a feature at the corner points of the domain, depends on the order of the singularity and depends on the order of the singularity and on the values of approach angles of an elliptic part of boundary of the domain to the line change of type. We have shown that for what angles of approach a singular solution of homogeneous Neumann-Tricomi problem belongs to the space L 2. In case of Neumann-Tricomi problem, unlike Tricomi problem in space L 2, only the value of angle at point A solves everything and the angle of approach at point B does not react []. Also, we have obtained conditions of existence of n-regular solutions for the nonhomogeneous Neumann-Tricomi problem. These conditions have been formulated in terms of orthogonality of the function in the right hand side of the equation to the corresponding singular solutions of its adjoint homogeneous problem. Acknowledgements. The authors are grateful to Professor T. Sh. Kalmenov for his support and attention to this work. This work has been supported by Committee of Science of Ministry of Education and Science of the Republic of Kazakhstan grant number 743. References [] A. V. Bitsadze; On the problem of equations of mixed type. Trudy Mat. Inst. Steklov., 4 953, pp [2] A. Erd elyi, W. Magnus, F. Oberherhettinger, F. G. Tricomi; Tablititsy integral nykh preobrazovaninii. T.. Preobrazovaniya Fur e, Laplasa, Mellina Tables of Integral Transforms. Vol.. Fourier, Laplace, Mellin Transforms. Moscow: Nauka, 969. [3] T. Sh. Kalmenov, A. B. Bazarbekov; A Criterion for the Strong Solvability of a Tricomi Problem for the Lavrent ev-bitsadze Equation. Dokl. Akad. Nauk SSSR, , pp [4] T. Sh. Kalmenov, A. B. Bazarbekov; A Criterion for the Strong Solvability of a Tricomi Problem for the Lavrent ev-bitsadze Equation in L p. Differential Equations, , pp
9 EJDE-24/93 SOLUTIONS TO NEUMANN-TRICOMI PROBLEMS 9 [5] T. Sh. Kalmenov, M. A. Sadybekov, N.E. Erzhanov; A Criterion for the Strong Solvability of the Tricomi Problem for the Lavrent ev-bitsadze Equation. The General Case. Differential Equations, , pp [6] M. A. Lavrent ev, B. V. Shabat; Metody therii funktsii kompleksnogo premennogo Methods of the Theory of Functions of a Complex Variable. Moscow, 967. [7] E. I. Moiseev, M. Mogimi; On the Completeness of Eigenfunctions of the Neumann-Tricomi Problem for a Degenerate Equation of Mixed Type. Differential Equations, , pp [8] E. I. Moiseev, P. V. Nefedov; Tricomi problem for the Lavrent ev Bitsadze equation in a 3D domain. Integral Transforms and Special Functions, 23 22, pp [9] L. S. Pulkina; Certain nonlocal problem for a degenerate hyperbolic equation. Mathematical Notes, , pp [] A. V. Rogovoy; Properties of solutions of the Tricomi problem for the Lavrent ev-bitsadze equation at corner points. Differential Equations, , pp [] M. A. Sadybekov, N. A. Yessirkegenov; Strong solvability criterion of Neumann-Tricomi problem in L 2 for the Lavrent ev-bitsadze equation. Mathematical Journal, 47 23, pp [2] M. M. Smirnov; Uravneniya smeshannogo tipa Equations of Mixed Type. Moscow: Vyssh. Shkola 985. Makhmud A. Sadybekov Institute of Mathematics and Mathematical Modeling, 25 Pushkin str., 5 Almaty, Kazakhstan address: makhmud-s@mail.ru Nurgissa A. Yessirkegenov Institute of Mathematics and Mathematical Modeling, 25 Pushkin str., 5 Almaty, Kazakhstan. Al-Farabi Kazakh national university, 7 ave. Al-Farabi, 54 Almaty, Kazakhstan address: nurgisa@hotmail.com
INITIAL-BOUNDARY VALUE PROBLEMS FOR THE WAVE EQUATION. 1. Introduction In Ω = (0, 1) consider the one-dimensional potential u(x) =
Electronic Journal of Differential Equations, Vol. 4 (4), No. 48, pp. 6. ISSN: 7-669. URL: http://ejde.math.tstate.edu or http://ejde.math.unt.edu ftp ejde.math.tstate.edu INITIAL-BOUNDARY VALUE PROBLEMS
More informationNON-EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION SYSTEM WITH NONLOCAL SOURCES
Electronic Journal of Differential Equations, Vol. 2016 (2016, No. 45, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NON-EXTINCTION OF
More informationON THE CORRECT FORMULATION OF A MULTIDIMENSIONAL PROBLEM FOR STRICTLY HYPERBOLIC EQUATIONS OF HIGHER ORDER
Georgian Mathematical Journal 1(1994), No., 141-150 ON THE CORRECT FORMULATION OF A MULTIDIMENSIONAL PROBLEM FOR STRICTLY HYPERBOLIC EQUATIONS OF HIGHER ORDER S. KHARIBEGASHVILI Abstract. A theorem of
More informationu xx + u yy = 0. (5.1)
Chapter 5 Laplace Equation The following equation is called Laplace equation in two independent variables x, y: The non-homogeneous problem u xx + u yy =. (5.1) u xx + u yy = F, (5.) where F is a function
More informationSufficient conditions for functions to form Riesz bases in L 2 and applications to nonlinear boundary-value problems
Electronic Journal of Differential Equations, Vol. 200(200), No. 74, pp. 0. ISSN: 072-669. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Sufficient conditions
More informationRIEMANN-HILBERT PROBLEM FOR THE MOISIL-TEODORESCU SYSTEM IN MULTIPLE CONNECTED DOMAINS
Electronic Journal of Differential Equations, Vol. 2016 (2016), No. 310, pp. 1 5. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu RIEMANN-HILBERT PROBLEM FOR THE MOISIL-TEODORESCU
More informationABOUT A METHOD OF RESEARCH OF THE NON-LOCAL PROBLEM FOR THE LOADED MIXED TYPE EQUATION IN DOUBLE-CONNECTED DOMAIN. O.Kh.
Bulletin KRASEC. Phys. & Math. Sci, 014, V. 9,., pp. 3-1. ISSN 313-0156 MSC 35M10 DOI: 10.18454/313-0156-014-9--3-1 ABOUT A METHOD OF RESEARCH OF THE NON-LOCAL PROBLEM FOR THE LOADED MIXED TYPE EQUATION
More informationSTABILITY OF BOUNDARY-VALUE PROBLEMS FOR THIRD-ORDER PARTIAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 217 (217), No. 53, pp. 1 11. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu STABILITY OF BOUNDARY-VALUE PROBLEMS FOR THIRD-ORDER
More informationSINC PACK, and Separation of Variables
SINC PACK, and Separation of Variables Frank Stenger Abstract This talk consists of a proof of part of Stenger s SINC-PACK computer package (an approx. 400-page tutorial + about 250 Matlab programs) that
More informationIsoperimetric Inequalities for the Cauchy-Dirichlet Heat Operator
Filomat 32:3 (218), 885 892 https://doi.org/1.2298/fil183885k Published by Faculty of Sciences and Mathematics, University of Niš, Serbia Available at: http://www.pmf.ni.ac.rs/filomat Isoperimetric Inequalities
More informationSEMILINEAR ELLIPTIC EQUATIONS WITH DEPENDENCE ON THE GRADIENT
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 139, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SEMILINEAR ELLIPTIC
More informationBOUNDARY PROPERTIES OF FIRST-ORDER PARTIAL DERIVATIVES OF THE POISSON INTEGRAL FOR THE HALF-SPACE R + k+1. (k > 1)
GEORGIAN MATHEMATICAL JOURNAL: Vol. 4, No. 6, 1997, 585-6 BOUNDARY PROPERTIES OF FIRST-ORDER PARTIAL DERIVATIVES OF THE POISSON INTEGRAL FOR THE HALF-SPACE R + k+1 (k > 1) S. TOPURIA Abstract. Boundary
More informationREPRESENTATION OF THE NORMING CONSTANTS BY TWO SPECTRA
Electronic Journal of Differential Equations, Vol. 0000, No. 59, pp. 0. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu REPRESENTATION OF THE NORMING
More informationGREEN S FUNCTIONAL FOR SECOND-ORDER LINEAR DIFFERENTIAL EQUATION WITH NONLOCAL CONDITIONS
Electronic Journal of Differential Equations, Vol. 1 (1), No. 11, pp. 1 1. ISSN: 17-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu GREEN S FUNCTIONAL FOR
More informationEXISTENCE AND UNIQUENESS FOR BOUNDARY-VALUE PROBLEM WITH ADDITIONAL SINGLE POINT CONDITIONS OF THE STOKES-BITSADZE SYSTEM
Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 202, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.tstate.edu or http://ejde.math.unt.edu ftp ejde.math.tstate.edu EXISTENCE AND UNIQUENESS
More informationElliptic Problems for Pseudo Differential Equations in a Polyhedral Cone
Advances in Dynamical Systems and Applications ISSN 0973-5321, Volume 9, Number 2, pp. 227 237 (2014) http://campus.mst.edu/adsa Elliptic Problems for Pseudo Differential Equations in a Polyhedral Cone
More informationSOLVABILITY IN THE SENSE OF SEQUENCES TO SOME NON-FREDHOLM OPERATORS. 1. Introduction
Electronic Journal of Differential Equations, Vol. 203 (203), No. 60, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SOLVABILITY IN THE SENSE
More informationPERIODIC PROBLEM FOR AN IMPULSIVE SYSTEM OF THE LOADED HYPERBOLIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 72, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu PERIODIC PROBLEM FOR AN IMPULSIVE SYSTEM
More informationMATH 452. SAMPLE 3 SOLUTIONS May 3, (10 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic.
MATH 45 SAMPLE 3 SOLUTIONS May 3, 06. (0 pts) Let f(x + iy) = u(x, y) + iv(x, y) be an analytic function. Show that u(x, y) is harmonic. Because f is holomorphic, u and v satisfy the Cauchy-Riemann equations:
More informationTRICOMI PROBLEM FOR THE ELLIPTIC-HYPERBOLIC EQUATION OF THE SECOND KIND. M.S. Salahitdinov and N.K. Mamadaliev
Korean J. Math. 19 (211), No. 2, pp. 111 127 TRICOMI PROBLEM FOR THE ELLIPTIC-HYPERBOLIC EQUATION OF THE SECOND KIND M.S. Salahitdinov and N.K. Mamadaliev Abstract. We prove the uniqueness solvability
More informationCLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE
CLASSIFICATION AND PRINCIPLE OF SUPERPOSITION FOR SECOND ORDER LINEAR PDE 1. Linear Partial Differential Equations A partial differential equation (PDE) is an equation, for an unknown function u, that
More informationSOLVABILITY OF A NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION WITH INTEGRAL CONDITIONS
Electronic Journal of Differential Equations, Vol. 27 27, No. 7, pp. 2. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu SOLVABILITY OF A NONLOCAL PROBLEM FOR A HYPERBOLIC EQUATION
More informationA COUNTEREXAMPLE TO AN ENDPOINT BILINEAR STRICHARTZ INEQUALITY TERENCE TAO. t L x (R R2 ) f L 2 x (R2 )
Electronic Journal of Differential Equations, Vol. 2006(2006), No. 5, pp. 6. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A COUNTEREXAMPLE
More informationLucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche
Scuola di Dottorato THE WAVE EQUATION Lucio Demeio Dipartimento di Ingegneria Industriale e delle Scienze Matematiche Lucio Demeio - DIISM wave equation 1 / 44 1 The Vibrating String Equation 2 Second
More informationON BASICITY OF A SYSTEM OF EXPONENTS WITH DEGENERATING COEFFICIENTS
TWMS J. Pure Appl. Math. V.1, N.2, 2010, pp. 257-264 ON BASICITY OF A SYSTEM OF EXPONENTS WITH DEGENERATING COEFFICIENTS S.G. VELIYEV 1, A.R.SAFAROVA 2 Abstract. A system of exponents of power character
More informationAn Approximate Solution for Volterra Integral Equations of the Second Kind in Space with Weight Function
International Journal of Mathematical Analysis Vol. 11 17 no. 18 849-861 HIKARI Ltd www.m-hikari.com https://doi.org/1.1988/ijma.17.771 An Approximate Solution for Volterra Integral Equations of the Second
More informationSOLVABILITY OF MULTIPOINT DIFFERENTIAL OPERATORS OF FIRST ORDER
Electronic Journal of Differential Equations, Vol. 2015 2015, No. 36, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu SOLVABILITY OF MULTIPOINT
More informationA non-local problem with integral conditions for hyperbolic equations
Electronic Journal of Differential Equations, Vol. 1999(1999), No. 45, pp. 1 6. ISSN: 17-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login:
More informationTHE DEPENDENCE OF SOLUTION UNIQUENESS CLASSES OF BOUNDARY VALUE PROBLEMS FOR GENERAL PARABOLIC SYSTEMS ON THE GEOMETRY OF AN UNBOUNDED DOMAIN
GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 4, 1998, 31-33 THE DEPENDENCE OF SOLUTION UNIQUENESS CLASSES OF BOUNDARY VALUE PROBLEMS FOR GENERAL PARABOLIC SYSTEMS ON THE GEOMETRY OF AN UNBOUNDED DOMAIN A.
More informationFinal Examination Solutions
Math. 42 Fulling) 4 December 25 Final Examination Solutions Calculators may be used for simple arithmetic operations only! Laplacian operator in polar coordinates: Some possibly useful information 2 u
More informationIn this chapter we study elliptical PDEs. That is, PDEs of the form. 2 u = lots,
Chapter 8 Elliptic PDEs In this chapter we study elliptical PDEs. That is, PDEs of the form 2 u = lots, where lots means lower-order terms (u x, u y,..., u, f). Here are some ways to think about the physical
More informationBOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES
1 BOUNDARY-VALUE PROBLEMS IN RECTANGULAR COORDINATES 1.1 Separable Partial Differential Equations 1. Classical PDEs and Boundary-Value Problems 1.3 Heat Equation 1.4 Wave Equation 1.5 Laplace s Equation
More informationSOURCE IDENTIFICATION FOR THE HEAT EQUATION WITH MEMORY. Sergei Avdonin, Gulden Murzabekova, and Karlygash Nurtazina. IMA Preprint Series #2459
SOURCE IDENTIFICATION FOR THE HEAT EQUATION WITH MEMORY By Sergei Avdonin, Gulden Murzabekova, and Karlygash Nurtazina IMA Preprint Series #2459 (November 215) INSTITUTE FOR MATHEMATICS AND ITS APPLICATIONS
More information+ (k > 1) S. TOPURIA. Abstract. The boundary properties of second-order partial derivatives of the Poisson integral are studied for a half-space R k+1
GEORGIAN MATHEMATICAL JOURNAL: Vol. 5, No. 4, 1998, 85-4 BOUNDARY PROPERTIES OF SECOND-ORDER PARTIAL DERIVATIVES OF THE POISSON INTEGRAL FOR A HALF-SPACE +1 + (k > 1) S. TOPURIA Abstract. The boundary
More informationMATH 220: Problem Set 3 Solutions
MATH 220: Problem Set 3 Solutions Problem 1. Let ψ C() be given by: 0, x < 1, 1 + x, 1 < x < 0, ψ(x) = 1 x, 0 < x < 1, 0, x > 1, so that it verifies ψ 0, ψ(x) = 0 if x 1 and ψ(x)dx = 1. Consider (ψ j )
More informationSalmon: Lectures on partial differential equations
6. The wave equation Of the 3 basic equations derived in the previous section, we have already discussed the heat equation, (1) θ t = κθ xx + Q( x,t). In this section we discuss the wave equation, () θ
More informationLIFE SPAN OF BLOW-UP SOLUTIONS FOR HIGHER-ORDER SEMILINEAR PARABOLIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 21(21), No. 17, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu LIFE SPAN OF BLOW-UP
More informationFinal: Solutions Math 118A, Fall 2013
Final: Solutions Math 118A, Fall 2013 1. [20 pts] For each of the following PDEs for u(x, y), give their order and say if they are nonlinear or linear. If they are linear, say if they are homogeneous or
More informationChapter 3 Second Order Linear Equations
Partial Differential Equations (Math 3303) A Ë@ Õæ Aë áöß @. X. @ 2015-2014 ú GA JË@ É Ë@ Chapter 3 Second Order Linear Equations Second-order partial differential equations for an known function u(x,
More informationGENERIC SOLVABILITY FOR THE 3-D NAVIER-STOKES EQUATIONS WITH NONREGULAR FORCE
Electronic Journal of Differential Equations, Vol. 2(2), No. 78, pp. 1 8. ISSN: 172-6691. UR: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) GENERIC SOVABIITY
More informationBanach Journal of Mathematical Analysis ISSN: (electronic)
Banach J. Math. Anal. 1 (2007), no. 1, 56 65 Banach Journal of Mathematical Analysis ISSN: 1735-8787 (electronic) http://www.math-analysis.org SOME REMARKS ON STABILITY AND SOLVABILITY OF LINEAR FUNCTIONAL
More informationMULTIPLE SOLUTIONS FOR CRITICAL ELLIPTIC PROBLEMS WITH FRACTIONAL LAPLACIAN
Electronic Journal of Differential Equations, Vol. 016 (016), No. 97, pp. 1 11. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE SOLUTIONS
More informationSELF-ADJOINTNESS OF SCHRÖDINGER-TYPE OPERATORS WITH SINGULAR POTENTIALS ON MANIFOLDS OF BOUNDED GEOMETRY
Electronic Journal of Differential Equations, Vol. 2003(2003), No.??, pp. 1 8. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) SELF-ADJOINTNESS
More informationEXISTENCE AND UNIQUENESS FOR A TWO-POINT INTERFACE BOUNDARY VALUE PROBLEM
Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 242, pp. 1 12. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE AND
More information21 Laplace s Equation and Harmonic Functions
2 Laplace s Equation and Harmonic Functions 2. Introductory Remarks on the Laplacian operator Given a domain Ω R d, then 2 u = div(grad u) = in Ω () is Laplace s equation defined in Ω. If d = 2, in cartesian
More informationFORMAL AND ANALYTIC SOLUTIONS FOR A QUADRIC ITERATIVE FUNCTIONAL EQUATION
Electronic Journal of Differential Equations, Vol. 202 (202), No. 46, pp. 9. ISSN: 072-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu FORMAL AND ANALYTIC SOLUTIONS
More informationOn the convergence rate of a difference solution of the Poisson equation with fully nonlocal constraints
Nonlinear Analysis: Modelling and Control, 04, Vol. 9, No. 3, 367 38 367 http://dx.doi.org/0.5388/na.04.3.4 On the convergence rate of a difference solution of the Poisson equation with fully nonlocal
More informationConformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder.
Lent 29 COMPLEX METHODS G. Taylor A star means optional and not necessarily harder. Conformal maps. (i) Let f(z) = az + b, with ad bc. Where in C is f conformal? cz + d (ii) Let f(z) = z +. What are the
More informationEXISTENCE OF SOLUTIONS FOR ONE-DIMENSIONAL WAVE EQUATIONS WITH NONLOCAL CONDITIONS
Eectronic Journa of Differentia Equations, Vo. 21(21), No. 76, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (ogin: ftp) EXISTENCE OF SOLUTIONS
More informationPartial Differential Equations for Engineering Math 312, Fall 2012
Partial Differential Equations for Engineering Math 312, Fall 2012 Jens Lorenz July 17, 2012 Contents Department of Mathematics and Statistics, UNM, Albuquerque, NM 87131 1 Second Order ODEs with Constant
More informationA PROPERTY OF SOBOLEV SPACES ON COMPLETE RIEMANNIAN MANIFOLDS
Electronic Journal of Differential Equations, Vol. 2005(2005), No.??, pp. 1 10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) A PROPERTY
More informationMATH-UA 263 Partial Differential Equations Recitation Summary
MATH-UA 263 Partial Differential Equations Recitation Summary Yuanxun (Bill) Bao Office Hour: Wednesday 2-4pm, WWH 1003 Email: yxb201@nyu.edu 1 February 2, 2018 Topics: verifying solution to a PDE, dispersion
More informationFirst order Partial Differential equations
First order Partial Differential equations 0.1 Introduction Definition 0.1.1 A Partial Deferential equation is called linear if the dependent variable and all its derivatives have degree one and not multiple
More informationMath 4263 Homework Set 1
Homework Set 1 1. Solve the following PDE/BVP 2. Solve the following PDE/BVP 2u t + 3u x = 0 u (x, 0) = sin (x) u x + e x u y = 0 u (0, y) = y 2 3. (a) Find the curves γ : t (x (t), y (t)) such that that
More informationPartial Differential Equations
M3M3 Partial Differential Equations Solutions to problem sheet 3/4 1* (i) Show that the second order linear differential operators L and M, defined in some domain Ω R n, and given by Mφ = Lφ = j=1 j=1
More informationComplex Representation in Two-Dimensional Theory of Elasticity
Complex Representation in Two-Dimensional Theory of Elasticity E. Vondenhoff 7-june-2006 Literature: Muskhelishvili : Some Basic Problems of the Mathematical Theory of Elasticity, Chapter 5 Prof. ir. C.
More informationHarmonic Analysis for Star Graphs and the Spherical Coordinate Trapezoidal Rule
Harmonic Analysis for Star Graphs and the Spherical Coordinate Trapezoidal Rule Robert Carlson University of Colorado at Colorado Springs September 24, 29 Abstract Aspects of continuous and discrete harmonic
More information6 Non-homogeneous Heat Problems
6 Non-homogeneous Heat Problems Up to this point all the problems we have considered for the heat or wave equation we what we call homogeneous problems. This means that for an interval < x < l the problems
More informationAnalysis IV : Assignment 3 Solutions John Toth, Winter ,...). In particular for every fixed m N the sequence (u (n)
Analysis IV : Assignment 3 Solutions John Toth, Winter 203 Exercise (l 2 (Z), 2 ) is a complete and separable Hilbert space. Proof Let {u (n) } n N be a Cauchy sequence. Say u (n) = (..., u 2, (n) u (n),
More informationEXISTENCE OF POSITIVE SOLUTIONS FOR KIRCHHOFF TYPE EQUATIONS
Electronic Journal of Differential Equations, Vol. 013 013, No. 180, pp. 1 8. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu EXISTENCE OF POSITIVE
More informationLinear Algebra. Session 12
Linear Algebra. Session 12 Dr. Marco A Roque Sol 08/01/2017 Example 12.1 Find the constant function that is the least squares fit to the following data x 0 1 2 3 f(x) 1 0 1 2 Solution c = 1 c = 0 f (x)
More informationINTRODUCTION TO PDEs
INTRODUCTION TO PDEs In this course we are interested in the numerical approximation of PDEs using finite difference methods (FDM). We will use some simple prototype boundary value problems (BVP) and initial
More informationINVERSE PROBLEMS OF PERIODIC SPATIAL DISTRIBUTIONS FOR A TIME FRACTIONAL DIFFUSION EQUATION
Electronic Journal of Differential Equations, Vol. 216 (216), No. 14, pp. 1 9. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu INVERSE PROBLEMS OF
More informationREGULARITY OF WEAK SOLUTIONS TO THE LANDAU-LIFSHITZ SYSTEM IN BOUNDED REGULAR DOMAINS
Electronic Journal of Differential Equations, Vol. 20072007, No. 141, pp. 1 9. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu login: ftp REGULARITY
More informationASYMPTOTIC THEORY FOR WEAKLY NON-LINEAR WAVE EQUATIONS IN SEMI-INFINITE DOMAINS
Electronic Journal of Differential Equations, Vol. 004(004), No. 07, pp. 8. ISSN: 07-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu (login: ftp) ASYMPTOTIC
More informationUNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH
UNIVERSITY of LIMERICK OLLSCOIL LUIMNIGH Faculty of Science and Engineering Department of Mathematics and Statistics END OF SEMESTER ASSESSMENT PAPER MODULE CODE: MA4006 SEMESTER: Spring 2011 MODULE TITLE:
More informationCauchy Problem for One Class of Ordinary Differential Equations
Int. Journal of Math. Analysis, Vol. 6, 212, no. 14, 695-699 Cauchy Problem for One Class of Ordinary Differential Equations A. Tungatarov Department of Mechanics and Mathematics Al-Farabi Kazakh National
More informationEFFECTIVE SOLUTIONS OF SOME DUAL INTEGRAL EQUATIONS AND THEIR APPLICATIONS
GENEALIZATIONS OF COMPLEX ANALYSIS BANACH CENTE PUBLICATIONS, VOLUME 37 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WASZAWA 996 EFFECTIVE SOLUTIONS OF SOME DUAL INTEGAL EQUATIONS AND THEI APPLICATIONS
More informationQualification Exam: Mathematical Methods
Qualification Exam: Mathematical Methods Name:, QEID#41534189: August, 218 Qualification Exam QEID#41534189 2 1 Mathematical Methods I Problem 1. ID:MM-1-2 Solve the differential equation dy + y = sin
More informationf(s) e -i n π s/l d s
Pointwise convergence of complex Fourier series Let f(x) be a periodic function with period l defined on the interval [,l]. The complex Fourier coefficients of f( x) are This leads to a Fourier series
More informationMULTIPLE POSITIVE SOLUTIONS FOR KIRCHHOFF PROBLEMS WITH SIGN-CHANGING POTENTIAL
Electronic Journal of Differential Equations, Vol. 015 015), No. 0, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu MULTIPLE POSITIVE SOLUTIONS
More informationON ISOMORPHISM OF TWO BASES IN MORREY-LEBESGUE TYPE SPACES
Sahand Communications in Mathematical Analysis (SCMA) Vol. 4 No. 1 (2016), 79-90 http://scma.maragheh.ac.ir ON ISOMORPHISM OF TWO BASES IN MORREY-LEBESGUE TYPE SPACES FATIMA A. GULIYEVA 1 AND RUBABA H.
More informationASYMPTOTIC PROPERTIES OF SOLUTIONS TO LINEAR NONAUTONOMOUS DELAY DIFFERENTIAL EQUATIONS THROUGH GENERALIZED CHARACTERISTIC EQUATIONS
Electronic Journal of Differential Equations, Vol. 2(2), No. 95, pp. 5. ISSN: 72-669. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu ASYMPTOTIC PROPERTIES OF SOLUTIONS
More informationPICONE S IDENTITY FOR A SYSTEM OF FIRST-ORDER NONLINEAR PARTIAL DIFFERENTIAL EQUATIONS
Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 143, pp. 1 7. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu PICONE S IDENTITY
More informationThe double layer potential
The double layer potential In this project, our goal is to explain how the Dirichlet problem for a linear elliptic partial differential equation can be converted into an integral equation by representing
More informationDifferential equations, comprehensive exam topics and sample questions
Differential equations, comprehensive exam topics and sample questions Topics covered ODE s: Chapters -5, 7, from Elementary Differential Equations by Edwards and Penney, 6th edition.. Exact solutions
More informationInstitute of Mathematics, Russian Academy of Sciences Universitetskiĭ Prosp. 4, Novosibirsk, Russia
PARTIAL DIFFERENTIAL EQUATIONS BANACH CENTER PUBLICATIONS, VOLUME 27 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 1992 L p -THEORY OF BOUNDARY VALUE PROBLEMS FOR SOBOLEV TYPE EQUATIONS
More informationHyperbolic PDEs. Chapter 6
Chapter 6 Hyperbolic PDEs In this chapter we will prove existence, uniqueness, and continuous dependence of solutions to hyperbolic PDEs in a variety of domains. To get a feel for what we might expect,
More informationDegree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m.
Degree Master of Science in Mathematical Modelling and Scientific Computing Mathematical Methods I Thursday, 12th January 2012, 9:30 a.m.- 11:30 a.m. Candidates should submit answers to a maximum of four
More informationIn this worksheet we will use the eigenfunction expansion to solve nonhomogeneous equation.
Eigen Function Expansion and Applications. In this worksheet we will use the eigenfunction expansion to solve nonhomogeneous equation. a/ The theory. b/ Example: Solving the Euler equation in two ways.
More information17 Source Problems for Heat and Wave IB- VPs
17 Source Problems for Heat and Wave IB- VPs We have mostly dealt with homogeneous equations, homogeneous b.c.s in this course so far. Recall that if we have non-homogeneous b.c.s, then we want to first
More informationInfinite Series. 1 Introduction. 2 General discussion on convergence
Infinite Series 1 Introduction I will only cover a few topics in this lecture, choosing to discuss those which I have used over the years. The text covers substantially more material and is available for
More informationPartial differential equation for temperature u(x, t) in a heat conducting insulated rod along the x-axis is given by the Heat equation:
Chapter 7 Heat Equation Partial differential equation for temperature u(x, t) in a heat conducting insulated rod along the x-axis is given by the Heat equation: u t = ku x x, x, t > (7.1) Here k is a constant
More informationMAT 372 K.T.D.D. Final Sınavın Çözümleri N. Course. Question 1 (Canonical Forms). Consider the second order partial differential equation
OKAN ÜNİVERSİTESİ FEN EDEBİYAT FAKÜTESİ MATEMATİK BÖÜMÜ 1.5. MAT 7 K.T.D.D. Final Sınavın Çözümleri N. Course Question 1 (Canonical Forms). Consider the second order partial differential equation (sin
More informationCOMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED TIME SCALES. Hüseyin Tuna
Indian J. Pure Appl. Math., 47(3): 535-544, September 2016 c Indian National Science Academy DOI: 10.1007/s13226-016-0196-1 COMPLETENESS THEOREM FOR THE DISSIPATIVE STURM-LIOUVILLE OPERATOR ON BOUNDED
More informationNONHOMOGENEOUS ELLIPTIC EQUATIONS INVOLVING CRITICAL SOBOLEV EXPONENT AND WEIGHT
Electronic Journal of Differential Equations, Vol. 016 (016), No. 08, pp. 1 1. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu NONHOMOGENEOUS ELLIPTIC
More informationBLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST DIFFUSION EQUATION WITH HOMOGENEOUS NEUMANN BOUNDARY CONDITIONS
Electronic Journal of Differential Equations, Vol. 016 (016), No. 36, pp. 1 10. ISSN: 107-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu BLOW-UP AND EXTINCTION OF SOLUTIONS TO A FAST
More informationLEGENDRE POLYNOMIALS AND APPLICATIONS. We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.
LEGENDRE POLYNOMIALS AND APPLICATIONS We construct Legendre polynomials and apply them to solve Dirichlet problems in spherical coordinates.. Legendre equation: series solutions The Legendre equation is
More informationand monotonicity property of these means is proved. The related mean value theorems of Cauchy type are also given.
Rad HAZU Volume 515 013, 1-10 ON LOGARITHMIC CONVEXITY FOR GIACCARDI S DIFFERENCE J PEČARIĆ AND ATIQ UR REHMAN Abstract In this paper, the Giaccardi s difference is considered in some special cases By
More informationCOEFFICIENTS OF SINGULARITIES FOR A SIMPLY SUPPORTED PLATE PROBLEMS IN PLANE SECTORS
Electronic Journal of Differential Equations, Vol. 213 (213), No. 238, pp. 1 8. ISSN: 172-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu ftp ejde.math.txstate.edu COEFFICIENTS OF SINGULARITIES
More informationTwo special equations: Bessel s and Legendre s equations. p Fourier-Bessel and Fourier-Legendre series. p
LECTURE 1 Table of Contents Two special equations: Bessel s and Legendre s equations. p. 259-268. Fourier-Bessel and Fourier-Legendre series. p. 453-460. Boundary value problems in other coordinate system.
More informationworked out from first principles by parameterizing the path, etc. If however C is a A path C is a simple closed path if and only if the starting point
III.c Green s Theorem As mentioned repeatedly, if F is not a gradient field then F dr must be worked out from first principles by parameterizing the path, etc. If however is a simple closed path in the
More informationMath 46, Applied Math (Spring 2009): Final
Math 46, Applied Math (Spring 2009): Final 3 hours, 80 points total, 9 questions worth varying numbers of points 1. [8 points] Find an approximate solution to the following initial-value problem which
More informationSynopsis of Complex Analysis. Ryan D. Reece
Synopsis of Complex Analysis Ryan D. Reece December 7, 2006 Chapter Complex Numbers. The Parts of a Complex Number A complex number, z, is an ordered pair of real numbers similar to the points in the real
More informationMATH 220: MIDTERM OCTOBER 29, 2015
MATH 22: MIDTERM OCTOBER 29, 25 This is a closed book, closed notes, no electronic devices exam. There are 5 problems. Solve Problems -3 and one of Problems 4 and 5. Write your solutions to problems and
More informationLECTURE 5: THE METHOD OF STATIONARY PHASE
LECTURE 5: THE METHOD OF STATIONARY PHASE Some notions.. A crash course on Fourier transform For j =,, n, j = x j. D j = i j. For any multi-index α = (α,, α n ) N n. α = α + + α n. α! = α! α n!. x α =
More informationA new class of pseudodifferential operators with mixed homogenities
A new class of pseudodifferential operators with mixed homogenities Po-Lam Yung University of Oxford Jan 20, 2014 Introduction Given a smooth distribution of hyperplanes on R N (or more generally on a
More informationPartial Differential Equations
Part II Partial Differential Equations Year 2015 2014 2013 2012 2011 2010 2009 2008 2007 2006 2005 2015 Paper 4, Section II 29E Partial Differential Equations 72 (a) Show that the Cauchy problem for u(x,
More informationMAIN ARTICLES. In present paper we consider the Neumann problem for the operator equation
Volume 14, 2010 1 MAIN ARTICLES THE NEUMANN PROBLEM FOR A DEGENERATE DIFFERENTIAL OPERATOR EQUATION Liparit Tepoyan Yerevan State University, Faculty of mathematics and mechanics Abstract. We consider
More informationECONOMETRIC AND GEOMETRIC ANALYSIS OF COBB-DOUGLAS AND CES PRODUCTION FUNCTIONS
ECONOMETRIC AND GEOMETRIC ANALYSIS OF COBB-DOUGLAS AND CES PRODUCTION FUNCTIONS M. Zakhirov M. Ulugbek National University of Uzbekistan Abstract Due to the fact that the sign of mixed second order partial
More information