In this paper, we obtain the best constant in the Lyapunov-type inequality for thirdorder linear differential equations under the non-conjugate boundary conditions by bounding the Green function of the same problem. In this direction, to the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for the non-conjugate boundary value problems. By using such inequalities, we obtain sharp lower bounds for the eigenvalues of corresponding equations.
{"title":"Lyapunov-type inequalities for third-order linear differential equations under the non-conjugate boundary conditions","authors":"M. Aktas, D. Çakmak","doi":"10.7153/dea-2018-10-14","DOIUrl":"https://doi.org/10.7153/dea-2018-10-14","url":null,"abstract":"In this paper, we obtain the best constant in the Lyapunov-type inequality for thirdorder linear differential equations under the non-conjugate boundary conditions by bounding the Green function of the same problem. In this direction, to the best of our knowledge, there is no paper dealing with Lyapunov-type inequalities for the non-conjugate boundary value problems. By using such inequalities, we obtain sharp lower bounds for the eigenvalues of corresponding equations.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"219 1","pages":"219-226"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79794096","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An approach for solving general second-order, linear, variable-coefficient ordinary differential equations in standard form under initial-value conditions is presented for the case of a specific constant-form relation between the two otherwise arbitrary coefficients. The resulting system of linear equations produces fundamental (or state transition) matrix elements used to create integraland closed-form solutions for both homogeneous and nonhomogeneous differential equation variants. Two example equations are chosen to illustrate application. A short discussion is presented on the comparison of the theoretical results for these examples with the corresponding symbolic integration outputs provided by several commercial programs which were seen, at times, to be long and unwieldy or even non-existent. Mathematics subject classification (2010): 34A30, 93C15.
{"title":"On a General Class of Second-Order, Linear, Ordinary Differential Equations Solvable as a System of First-Order Equations","authors":"R. Pascone","doi":"10.7153/DEA-2018-10-08","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-08","url":null,"abstract":"An approach for solving general second-order, linear, variable-coefficient ordinary differential equations in standard form under initial-value conditions is presented for the case of a specific constant-form relation between the two otherwise arbitrary coefficients. The resulting system of linear equations produces fundamental (or state transition) matrix elements used to create integraland closed-form solutions for both homogeneous and nonhomogeneous differential equation variants. Two example equations are chosen to illustrate application. A short discussion is presented on the comparison of the theoretical results for these examples with the corresponding symbolic integration outputs provided by several commercial programs which were seen, at times, to be long and unwieldy or even non-existent. Mathematics subject classification (2010): 34A30, 93C15.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"8 1","pages":"131-146"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84220795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We observe that a version of Poincaré’s inequality for positive solutions to second order linear non-divergence form equations vanishing on a portion of the boundary, implies a natural connection between Lp Dirichlet and Lq Regularity problems for this type of equations.
{"title":"A connection between regularity and Dirichlet problems for non-divergence elliptic equations","authors":"J. Rivera-Noriega","doi":"10.7153/DEA-2018-10-05","DOIUrl":"https://doi.org/10.7153/DEA-2018-10-05","url":null,"abstract":"We observe that a version of Poincaré’s inequality for positive solutions to second order linear non-divergence form equations vanishing on a portion of the boundary, implies a natural connection between Lp Dirichlet and Lq Regularity problems for this type of equations.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"64 1","pages":"75-86"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89136342","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider establishing lower bounds for the first zero of the solution of the nonlinear second order initial value problem (p(x)y′(x))′ + f (x,y(x)) = 0, x 0 y(0) = a > 0, y′(0) = 0. Using the linear case as a starting point, we prove several of these theorems, comparing them by considering several examples. Mathematics subject classification (2010): 34C10, 34A34, 34A36.
考虑建立非线性二阶初值问题(p(x)y ' (x)) ' + f (x,y(x)) = 0, x 0 y(0) = a > 0, y '(0) = 0)解的第一个零的下界。以线性情况为出发点，我们证明了其中几个定理，并通过考虑几个例子对它们进行了比较。数学学科分类(2010):34C10, 34A34, 34A36。
{"title":"Lower bounds for the first zero for nonlinear second order differential equations","authors":"D. Biles","doi":"10.7153/dea-2018-10-13","DOIUrl":"https://doi.org/10.7153/dea-2018-10-13","url":null,"abstract":"We consider establishing lower bounds for the first zero of the solution of the nonlinear second order initial value problem (p(x)y′(x))′ + f (x,y(x)) = 0, x 0 y(0) = a > 0, y′(0) = 0. Using the linear case as a starting point, we prove several of these theorems, comparing them by considering several examples. Mathematics subject classification (2010): 34C10, 34A34, 34A36.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"15 3 1","pages":"209-218"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75796627","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we investigate the following fractional p -Laplacian problem ⎨⎩ (−Δ)pu = λ |u|p−2u+ |u| ps,α−2u |x|α in Ω, u = 0 on ∂Ω, where Ω is a bounded domain containing the origin in RN with Lipschitz boundary, p ∈ (1,∞) , s ∈ (0,1) , 0 α < ps < N and p∗s,α = (N −α)p/(N − ps) is the fractional Hardy-Sobolev exponent. We prove the existence, multiplicity and bifurcation results for the above problem. Our results extend some results in the literature for the fractional p -Laplacian problem involving critical Sobolev exponent and the p -Laplacian problem involving Hardy-Sobolev exponents.
{"title":"Existence and multiplicity results for the fractional p-Laplacian equation with Hardy-Sobolev exponents","authors":"Gai ia Ning, Zhiyong Wang, Jihui Zhang","doi":"10.7153/dea-2018-10-06","DOIUrl":"https://doi.org/10.7153/dea-2018-10-06","url":null,"abstract":"In this paper, we investigate the following fractional p -Laplacian problem ⎨⎩ (−Δ)pu = λ |u|p−2u+ |u| ps,α−2u |x|α in Ω, u = 0 on ∂Ω, where Ω is a bounded domain containing the origin in RN with Lipschitz boundary, p ∈ (1,∞) , s ∈ (0,1) , 0 α < ps < N and p∗s,α = (N −α)p/(N − ps) is the fractional Hardy-Sobolev exponent. We prove the existence, multiplicity and bifurcation results for the above problem. Our results extend some results in the literature for the fractional p -Laplacian problem involving critical Sobolev exponent and the p -Laplacian problem involving Hardy-Sobolev exponents.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"26 1","pages":"87-114"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"78741948","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
. We provide two existence results for sign-changing solutions to the Dirichlet problem for the family of equations − ∆ p u − ∆ q u = α | u | p − 2 u + β | u | q − 2 u , where 1 < q < p and α , β are parameters. First, we show the existence in the resonant case α ∈ σ ( − ∆ p ) for sufficiently large β , thereby generalizing previously known results. The obtained solutions have negative energy. Second, we show the existence for any β > λ 1 ( q ) and sufficiently large α under an additional nonresonant assumption, where λ 1 ( q ) is the first eigenvalue of the q -Laplacian. The obtained solutions have positive energy.
。给出了一类方程(-∆p u -∆qu = α | u | p - 2u + β | u | q - 2u)的Dirichlet问题变符号解的两个存在性结果，其中1 < q < p和α， β为参数。首先，我们证明了足够大的β在共振情况下α∈σ(−∆p)的存在性，从而推广了先前已知的结果。得到的解具有负能量。其次，在一个附加的非共振假设下，我们证明了任意β > λ 1 (q)和足够大的α的存在性，其中λ 1 (q)是q -拉普拉斯算子的第一特征值。得到的解具有正能量。
{"title":"On sign-changing solutions for resonant (p,q)-Laplace equations","authors":"V. Bobkov, Mieko Tanaka","doi":"10.7153/dea-2018-10-12","DOIUrl":"https://doi.org/10.7153/dea-2018-10-12","url":null,"abstract":". We provide two existence results for sign-changing solutions to the Dirichlet problem for the family of equations − ∆ p u − ∆ q u = α | u | p − 2 u + β | u | q − 2 u , where 1 < q < p and α , β are parameters. First, we show the existence in the resonant case α ∈ σ ( − ∆ p ) for sufficiently large β , thereby generalizing previously known results. The obtained solutions have negative energy. Second, we show the existence for any β > λ 1 ( q ) and sufficiently large α under an additional nonresonant assumption, where λ 1 ( q ) is the first eigenvalue of the q -Laplacian. The obtained solutions have positive energy.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"1 1","pages":"197-208"},"PeriodicalIF":0.0,"publicationDate":"2018-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83569650","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Norodin A. Rangaig, Norhamida D. Minor, G. F. Penonal, Jae Lord Dexter C. Filipinas, V. Convicto
A new Integral Transform was introduced in this paper. Fundamental properties of this transform were derived and presented such as the convolution identity, and step Heaviside function. It is proven and tested to solve some basic linear-differential equations and had succesfully solved the Abel's Generalized equation and derived the Volterra Integral Equation of the second kind by means of Initial Value Problem. The Natural Logarithm (e.g logex=lnx) has been established and defined by means of modifying the Euler Definite Integral based on the Rangaig's fomulation. Hence, this transform may solve some different kind of integral and differential equations and it competes with other known transforms like Laplace, Sumudu and Elzaki Transform. Keywords: Rangaig Transform, Integral Transform, linear ordinary differential function, Integro-differential equation, Convolution Theorem.
{"title":"On Another Type of Transform Called Rangaig Transform","authors":"Norodin A. Rangaig, Norhamida D. Minor, G. F. Penonal, Jae Lord Dexter C. Filipinas, V. Convicto","doi":"10.12691/IJPDEA-5-1-6","DOIUrl":"https://doi.org/10.12691/IJPDEA-5-1-6","url":null,"abstract":"A new Integral Transform was introduced in this paper. Fundamental properties of this transform were derived and presented such as the convolution identity, and step Heaviside function. It is proven and tested to solve some basic linear-differential equations and had succesfully solved the Abel's Generalized equation and derived the Volterra Integral Equation of the second kind by means of Initial Value Problem. The Natural Logarithm (e.g logex=lnx) has been established and defined by means of modifying the Euler Definite Integral based on the Rangaig's fomulation. Hence, this transform may solve some different kind of integral and differential equations and it competes with other known transforms like Laplace, Sumudu and Elzaki Transform. Keywords: Rangaig Transform, Integral Transform, linear ordinary differential function, Integro-differential equation, Convolution Theorem.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"5 1","pages":"42-48"},"PeriodicalIF":0.0,"publicationDate":"2017-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"85401441","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We studied optimal investment strategies for a plan contributor in a defined pension scheme, with stochastic salary and extra contributions, under the affine interest rate model. We considered two cases; where the extra contribution rates are stochastic and constant. We considered investment in three different assets namely risk free asset (cash), zero coupon bonds and the risky asset (stock). Using Legendre transformation method and dual theory, we obtained the optimal investment strategies the three investments using exponential utility function for the two cases. The result shows that the strategies for the respective investments used when there is no extra contribution can be used when the extra contribution rate is constant as in [1] but cannot be used when it is stochastic. Clearly this gives the member and the fund manager good insight on how to invest to maximize profit with minimal risk once this condition arises.
{"title":"Optimization of Wealth Investment Strategies for a DC Pension Fund with Stochastic Salary and Extra Contributions","authors":"E. Akpanibah, B. Osu, C. NjokuK.N., Eyo O. Akak","doi":"10.12691/IJPDEA-5-1-5","DOIUrl":"https://doi.org/10.12691/IJPDEA-5-1-5","url":null,"abstract":"We studied optimal investment strategies for a plan contributor in a defined pension scheme, with stochastic salary and extra contributions, under the affine interest rate model. We considered two cases; where the extra contribution rates are stochastic and constant. We considered investment in three different assets namely risk free asset (cash), zero coupon bonds and the risky asset (stock). Using Legendre transformation method and dual theory, we obtained the optimal investment strategies the three investments using exponential utility function for the two cases. The result shows that the strategies for the respective investments used when there is no extra contribution can be used when the extra contribution rate is constant as in [1] but cannot be used when it is stochastic. Clearly this gives the member and the fund manager good insight on how to invest to maximize profit with minimal risk once this condition arises.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"9 1","pages":"33-41"},"PeriodicalIF":0.0,"publicationDate":"2017-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"74651048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, the bi-quintic B-spline base functions are modified on a general 2-dimensional problem and then they are applied to two-dimensional Diffusion problem in order to obtain its numerical solutions. The computed results are compared with the results given in the literature.
{"title":"The Modified Bi-quintic B-spline Base Functions: An Application to Diffusion Equation","authors":"S. Kutluay, N. Yağmurlu","doi":"10.12691/IJPDEA-5-1-4","DOIUrl":"https://doi.org/10.12691/IJPDEA-5-1-4","url":null,"abstract":"In this paper, the bi-quintic B-spline base functions are modified on a general 2-dimensional problem and then they are applied to two-dimensional Diffusion problem in order to obtain its numerical solutions. The computed results are compared with the results given in the literature.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"43 1","pages":"26-32"},"PeriodicalIF":0.0,"publicationDate":"2017-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89961062","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The boundary element method (BEM) is a very effective numerical tool which has been widely applied in engineering problems. Wave equation is a very important equation in applied mathematics with many applications such as wave propagation analysis, acoustics, dynamics, health monitoring and etc. This paper presents to solve the nonlinear 2-D wave equation defined over a rectangular spatial domain with appropriate initial and boundary conditions. Numerical solutions of the governing equations are obtained by using the dual reciprocity boundary element method (DRBEM). Two-dimension wave equation is a time-domain problem, with three independent variables . At the first the Laplace transform is used to reduce by one the number of independent variables (in the present work ), then Salzer's method which is an effective numerical Laplace transform inversion algorithm is used to recover the solution of the original equation at time domain. The present method has been successfully applied to 2-D wave equation with satisfactory accuracy.
{"title":"Solving the Nonlinear Two-Dimension Wave Equation Using Dual Reciprocity Boundary Element Method","authors":"Kumars Mahmoodi, H. Ghassemi, A. Heydarian","doi":"10.12691/IJPDEA-5-1-3","DOIUrl":"https://doi.org/10.12691/IJPDEA-5-1-3","url":null,"abstract":"The boundary element method (BEM) is a very effective numerical tool which has been widely applied in engineering problems. Wave equation is a very important equation in applied mathematics with many applications such as wave propagation analysis, acoustics, dynamics, health monitoring and etc. This paper presents to solve the nonlinear 2-D wave equation defined over a rectangular spatial domain with appropriate initial and boundary conditions. Numerical solutions of the governing equations are obtained by using the dual reciprocity boundary element method (DRBEM). Two-dimension wave equation is a time-domain problem, with three independent variables . At the first the Laplace transform is used to reduce by one the number of independent variables (in the present work ), then Salzer's method which is an effective numerical Laplace transform inversion algorithm is used to recover the solution of the original equation at time domain. The present method has been successfully applied to 2-D wave equation with satisfactory accuracy.","PeriodicalId":11162,"journal":{"name":"Differential Equations and Applications","volume":"30 1","pages":"19-25"},"PeriodicalIF":0.0,"publicationDate":"2017-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"73550089","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：