In this paper, Chebyshev wavelets method is designed and presented to approximate the two-dimensional (2D) telegraph equation of hyperbolic type. Firstly, we transform the original problem into the equivalent integro-partial differential equations (integro-PDEs) containing the initial and boundary conditions. Thereafter, the operational matrices of integration and derivative of Chebyshev wavelets reduce the integro-PDEs into generalized Sylvester equations which are solved by Krylov subspace iterative method. The convergence analysis associated to the function approximation, and error estimation are also investigated. The presented procedure is found to be very effective and accurate, which is confirmed by the outcome of three test examples. The outcome of numerical results of errors achieved by the presented method is compared with some earlier works. The technique can be applied for higher dimension problems also, which is one of the key features of this technique.
{"title":"An Efficient Computational Approach Based on Chebyshev Wavelets for Two-Dimensional Hyperbolic Telegraph Equation With Convergence Analysis","authors":"Somveer Singh, Harendra Singh, Ramta Ram Pathak","doi":"10.1002/mma.70210","DOIUrl":"https://doi.org/10.1002/mma.70210","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, Chebyshev wavelets method is designed and presented to approximate the two-dimensional (2D) telegraph equation of hyperbolic type. Firstly, we transform the original problem into the equivalent integro-partial differential equations (integro-PDEs) containing the initial and boundary conditions. Thereafter, the operational matrices of integration and derivative of Chebyshev wavelets reduce the integro-PDEs into generalized Sylvester equations which are solved by Krylov subspace iterative method. The convergence analysis associated to the function approximation, and error estimation are also investigated. The presented procedure is found to be very effective and accurate, which is confirmed by the outcome of three test examples. The outcome of numerical results of errors achieved by the presented method is compared with some earlier works. The technique can be applied for higher dimension problems also, which is one of the key features of this technique.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 2","pages":"1036-1051"},"PeriodicalIF":1.8,"publicationDate":"2025-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145772548","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<div> <p>In this paper, the set of trajectories of the control system described by Urysohn type integral equation is considered. It is assumed that the system is nonlinear with respect to the state vector and affine with respect to the control vector. The closed ball of the space <span></span><math> <semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>p</mi> </mrow> </msub> <mo>,</mo> <mspace></mspace> <mi>p</mi> <mo>∈</mo> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mi>∞</mi> <mo>)</mo> </mrow> <annotation>$$ {L}_p,kern0.3em pin left(1,infty right) $$</annotation> </semantics></math>, is chosen as the set of admissible control functions. The trajectory of the system is defined as multivariable Lebesgue measurable function from the space <span></span><math> <semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>∞</mi> </mrow> </msub> </mrow> <annotation>$$ {L}_{infty } $$</annotation> </semantics></math> that satisfies the system's equation almost everywhere. Boundedness of the set of trajectories is shown, and it is proved that every sequence of trajectories has a subsequence that converges almost everywhere to a system's trajectory. Existence of the optimal process in the optimal control problem with linear quality functional is presented. It is shown that every trajectory is robust with respect to the fast consumption of the remaining control resource and the set of trajectories as a set valued map depending on <span></span><math> <semantics> <mrow> <mi>p</mi> </mrow> <annotation>$$ p $$</annotation> </semantics></math> is continuous with respect to <span></span><math> <semantics> <mrow> <mi>p</mi> </mrow> <annotation>$$ p $$</annotation> </semantics></math> in the Hausdorff pseudometric generated by the norm of the space <span></span><math> <semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>∞</mi> </mrow> </msub> </mrow> <annotation>$$ {L}_{infty } $$</annotation> </semant
本文考虑了用Urysohn型积分方程描述的控制系统的轨迹集。假设系统相对于状态向量是非线性的，相对于控制向量是仿射的。空间L p的闭球，p∈(1,∞)$$ {L}_p,kern0.3em pin left(1,infty right) $$，作为允许控制函数的集合。将系统的轨迹定义为空间L∞$$ {L}_{infty } $$上几乎处处满足系统方程的多变量勒贝格可测函数。给出了轨迹集的有界性，并证明了每一个轨迹序列都有一个几乎处处收敛于系统轨迹的子序列。给出了具有线性质量泛函的最优控制问题中最优过程的存在性。在空间范数生成的Hausdorff伪度量中，每个轨迹对于剩余控制资源的快速消耗都是鲁棒的，并且轨迹集作为依赖于p $$ p $$的集值映射相对于p $$ p $$是连续的L∞$$ {L}_{infty } $$。
{"title":"Properties of the Set of L∞ Trajectories of the Control Systems With Limited Control Resources","authors":"Nesir Huseyin, Anar Huseyin, Khalik G. Guseinov","doi":"10.1002/mma.70213","DOIUrl":"https://doi.org/10.1002/mma.70213","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, the set of trajectories of the control system described by Urysohn type integral equation is considered. It is assumed that the system is nonlinear with respect to the state vector and affine with respect to the control vector. The closed ball of the space \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>,</mo>\u0000 <mspace></mspace>\u0000 <mi>p</mi>\u0000 <mo>∈</mo>\u0000 <mo>(</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>∞</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ {L}_p,kern0.3em pin left(1,infty right) $$</annotation>\u0000 </semantics></math>, is chosen as the set of admissible control functions. The trajectory of the system is defined as multivariable Lebesgue measurable function from the space \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {L}_{infty } $$</annotation>\u0000 </semantics></math> that satisfies the system's equation almost everywhere. Boundedness of the set of trajectories is shown, and it is proved that every sequence of trajectories has a subsequence that converges almost everywhere to a system's trajectory. Existence of the optimal process in the optimal control problem with linear quality functional is presented. It is shown that every trajectory is robust with respect to the fast consumption of the remaining control resource and the set of trajectories as a set valued map depending on \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 <annotation>$$ p $$</annotation>\u0000 </semantics></math> is continuous with respect to \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 <annotation>$$ p $$</annotation>\u0000 </semantics></math> in the Hausdorff pseudometric generated by the norm of the space \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>∞</mi>\u0000 </mrow>\u0000 </msub>\u0000 </mrow>\u0000 <annotation>$$ {L}_{infty } $$</annotation>\u0000 </semant","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 2","pages":"1087-1096"},"PeriodicalIF":1.8,"publicationDate":"2025-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145772547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we study the normalized solutions of a quasilinear elliptic Choquard equation with critical Sobolev exponent and a mixed diffusion-type operator. The study begins by demonstrating the Hölder regularity of a weak solution, followed by existence results based on the exponent of the subcritical Choquard term.
{"title":"Normalized Solutions to a Quasilinear Equation Involving Critical Sobolev Exponent","authors":"Nidhi Nidhi, Konijeti Sreenadh","doi":"10.1002/mma.70212","DOIUrl":"https://doi.org/10.1002/mma.70212","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we study the normalized solutions of a quasilinear elliptic Choquard equation with critical Sobolev exponent and a mixed diffusion-type operator. The study begins by demonstrating the Hölder regularity of a weak solution, followed by existence results based on the exponent of the subcritical Choquard term.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 2","pages":"1070-1086"},"PeriodicalIF":1.8,"publicationDate":"2025-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145766384","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we deal with a numerical approximation of the solutions of systems involving the Caputo -shifted fractional derivatives. Convergence results are established in graded meshes. The convergence order of the numerical scheme is optimal in the sense that it does not depend on the scaling function . Various examples and numerical tests are performed to illustrate the efficiency of the proposed method.
{"title":"Convergence Study of a ψ-Shifted Fractional Differential Scheme","authors":"Fatma Jerbi","doi":"10.1002/mma.70227","DOIUrl":"https://doi.org/10.1002/mma.70227","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we deal with a numerical approximation of the solutions of systems involving the Caputo \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 </mrow>\u0000 <annotation>$$ psi $$</annotation>\u0000 </semantics></math>-shifted fractional derivatives. Convergence results are established in graded meshes. The convergence order of the numerical scheme is optimal in the sense that it does not depend on the scaling function \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 </mrow>\u0000 <annotation>$$ psi $$</annotation>\u0000 </semantics></math>. Various examples and numerical tests are performed to illustrate the efficiency of the proposed method.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 3","pages":"2076-2081"},"PeriodicalIF":1.8,"publicationDate":"2025-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145987131","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we consider a nonlinear eigenvalue problem consisting of nonlinear Sturm–Liouville equation with Dirichlet boundary conditions on a symmetric interval, where is the eigenparameter. We study the inverse problem associated with this problem in -framework and provide a procedure for constructing the nonlinear function .
在本文中，我们考虑由非线性Sturm-Liouville方程- y ' ' + k (x) f (y)组成的非线性特征值问题) = λ y $$ -{y}&amp;amp;amp;#x0005E;{prime prime }&amp;amp;amp;#x0002B;k(x)f(y)&amp;amp;amp;#x0003D;lambda y $$，在对称区间上具有Dirichlet边界条件，其中λ ＆gt； 0 $$ lambda &amp;amp;gt;0 $$为特征参数。我们在lq $$ {L}&amp;amp;amp;#x0005E;q $$ -框架中研究了与此问题相关的逆问题，并给出了构造非线性函数f (y)的一个过程。$$ f(y) $$。
{"title":"Lq-Inverse Problem for Nonlinear Sturm–Liouville Operator","authors":"Seyfollah Mosazadeh, Fatemeh Kiyaee","doi":"10.1002/mma.70152","DOIUrl":"https://doi.org/10.1002/mma.70152","url":null,"abstract":"<div>\u0000 \u0000 <p>In this paper, we consider a nonlinear eigenvalue problem consisting of nonlinear Sturm–Liouville equation \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>y</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>′</mo>\u0000 <mo>′</mo>\u0000 </mrow>\u0000 </msup>\u0000 <mo>+</mo>\u0000 <mi>k</mi>\u0000 <mo>(</mo>\u0000 <mi>x</mi>\u0000 <mo>)</mo>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>y</mi>\u0000 <mo>)</mo>\u0000 <mo>=</mo>\u0000 <mi>λ</mi>\u0000 <mi>y</mi>\u0000 </mrow>\u0000 <annotation>$$ -{y}&amp;amp;amp;amp;#x0005E;{prime prime }&amp;amp;amp;amp;#x0002B;k(x)f(y)&amp;amp;amp;amp;#x0003D;lambda y $$</annotation>\u0000 </semantics></math> with Dirichlet boundary conditions on a symmetric interval, where \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>λ</mi>\u0000 <mo>></mo>\u0000 <mn>0</mn>\u0000 </mrow>\u0000 <annotation>$$ lambda &amp;amp;amp;gt;0 $$</annotation>\u0000 </semantics></math> is the eigenparameter. We study the inverse problem associated with this problem in \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msup>\u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </msup>\u0000 </mrow>\u0000 <annotation>$$ {L}&amp;amp;amp;amp;#x0005E;q $$</annotation>\u0000 </semantics></math>-framework and provide a procedure for constructing the nonlinear function \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>(</mo>\u0000 <mi>y</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ f(y) $$</annotation>\u0000 </semantics></math>.</p>\u0000 </div>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 1","pages":"296-303"},"PeriodicalIF":1.8,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145750742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
<div> <p>Here, we will explore the equivalent integral equality (EIE) for instantaneous impulsive Hadamard fractional order system (IIHFrOS) and noninstantaneous impulsive Hadamard fractional order system (NIHFrOS). For one IIHFrOS, we apply the fractional property of segment function to construct the EIE of a special case of the IIHFrOS to deduce that the proposed EIE of the IIHFrOS in cited paper is incomplete. We combine the limit analysis with the limit properties of the IIHFrOS to find the correct IIHFrOS's EIE, and obtain another IIHFrOS's EIE by the connection between the two IIHFrOSs, which both IIHFrOSs' EIEs are an integral equation of the combination of <span></span><math> <semantics> <mrow> <mi>ψ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <annotation>$$ psi (t) $$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <msub> <mrow> <mi>Ψ</mi> </mrow> <mrow> <mi>j</mi> </mrow> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <annotation>$$ {Psi}_j(t) $$</annotation> </semantics></math> ( <span></span><math> <semantics> <mrow> <mi>j</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mi>…</mi> <mo>,</mo> <mi>N</mi> </mrow> <annotation>$$ j&amp;#x0003D;1,2,dots, N $$</annotation> </semantics></math>) with an arbitrary real. Next, we deduce the EIE of one NIHFrOS by the corresponding IIHFrOS and obtain another NIHFrOS's EIE by the connection between the two NIHFrOSs, which both NIHFrOSs' EIEs are an integral equation of the combination of the similar items of <span></span><math> <semantics> <mrow> <mi>ψ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <annotation>$$ psi (t) $$</annotation> </semantics></math> and <span></span><math> <semantics> <mrow> <msub> <mrow> <mi>Ψ</mi> </mrow> <mrow> <mi>j</mi> </mrow> </msub> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> <annotation>$$ {Psi}_j(t) $$</annotation> </sema
{"title":"Instantaneous and Noninstantaneous Impulsive Hadamard Fractional Order System and Their Equivalent Integral Equalities","authors":"Xianmin Zhang","doi":"10.1002/mma.70154","DOIUrl":"https://doi.org/10.1002/mma.70154","url":null,"abstract":"<div>\u0000 \u0000 <p>Here, we will explore the equivalent integral equality (EIE) for instantaneous impulsive Hadamard fractional order system (IIHFrOS) and noninstantaneous impulsive Hadamard fractional order system (NIHFrOS). For one IIHFrOS, we apply the fractional property of segment function to construct the EIE of a special case of the IIHFrOS to deduce that the proposed EIE of the IIHFrOS in cited paper is incomplete. We combine the limit analysis with the limit properties of the IIHFrOS to find the correct IIHFrOS's EIE, and obtain another IIHFrOS's EIE by the connection between the two IIHFrOSs, which both IIHFrOSs' EIEs are an integral equation of the combination of \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ psi (t) $$</annotation>\u0000 </semantics></math> and \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>Ψ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>j</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ {Psi}_j(t) $$</annotation>\u0000 </semantics></math> (\u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>j</mi>\u0000 <mo>=</mo>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mn>2</mn>\u0000 <mo>,</mo>\u0000 <mi>…</mi>\u0000 <mo>,</mo>\u0000 <mi>N</mi>\u0000 </mrow>\u0000 <annotation>$$ j&amp;amp;#x0003D;1,2,dots, N $$</annotation>\u0000 </semantics></math>) with an arbitrary real. Next, we deduce the EIE of one NIHFrOS by the corresponding IIHFrOS and obtain another NIHFrOS's EIE by the connection between the two NIHFrOSs, which both NIHFrOSs' EIEs are an integral equation of the combination of the similar items of \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>ψ</mi>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ psi (t) $$</annotation>\u0000 </semantics></math> and \u0000<span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <msub>\u0000 <mrow>\u0000 <mi>Ψ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>j</mi>\u0000 </mrow>\u0000 </msub>\u0000 <mo>(</mo>\u0000 <mi>t</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <annotation>$$ {Psi}_j(t) $$</annotation>\u0000 </sema","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 1","pages":"321-338"},"PeriodicalIF":1.8,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145739680","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Mokhtar Kirane, Ahmad Z. Fino, Berikbol T. Torebek, Zineb Sabbagh
This paper investigates the Fujita critical exponent for a heat equation with nonlinear memory posed on the Heisenberg groups. A sharp threshold is identified such that, for exponent values less than or equal to this critical value, no global solution exists, regardless of the choice of nonnegative initial data. Conversely, for exponent values above the threshold, global positive solutions exist under small initial data conditions. These findings extend the work of Cazenave, Dickstein, and Weissler, which addressed similar problems in the Euclidean setting. Furthermore, the paper provides lifespan estimates for local solutions under various initial data conditions. The analysis relies on the test function method and the Banach fixed-point theorem to establish the main results.
{"title":"Cazenave-Dickstein-Weissler-Type Extension of Fujita'S Problem on Heisenberg Groups","authors":"Mokhtar Kirane, Ahmad Z. Fino, Berikbol T. Torebek, Zineb Sabbagh","doi":"10.1002/mma.70166","DOIUrl":"https://doi.org/10.1002/mma.70166","url":null,"abstract":"<p>This paper investigates the Fujita critical exponent for a heat equation with nonlinear memory posed on the Heisenberg groups. A sharp threshold is identified such that, for exponent values less than or equal to this critical value, no global solution exists, regardless of the choice of nonnegative initial data. Conversely, for exponent values above the threshold, global positive solutions exist under small initial data conditions. These findings extend the work of Cazenave, Dickstein, and Weissler, which addressed similar problems in the Euclidean setting. Furthermore, the paper provides lifespan estimates for local solutions under various initial data conditions. The analysis relies on the test function method and the Banach fixed-point theorem to establish the main results.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 2","pages":"499-511"},"PeriodicalIF":1.8,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mma.70166","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145772509","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Dmitri V. Alexandrov, Eugenya V. Makoveeva, Alexandra E. Glebova, Liubov V. Toropova
A moving-boundary problem of synchronous directional and volumetric solidification in undercooled liquid drops is formulated and solved. Directional solidification is caused by the temperature gradient along the spatial direction, whereas volumetric growth of new phase elements is caused by the melt undercooling ahead of the phase transition boundary. As this takes place, the solidification domain is divided into three regions occupied by solid material, two-phase layer, and liquid phase. The heat and mass transfer model is formulated inside all regions with the corresponding boundary conditions. To solve the problem, we use self-similar variables and functions, as well as the Laplace method for evaluating the Laplace-type integral and the technique of small parameter expansion of unknown functions in series. The temperature, impurity concentration, solid phase fraction distributions, and laws of motion of two solidification boundaries are found analytically. Our approximate solution shows that nucleation and growth of particles ahead of the solid phase: two - phase region boundary leads to a U-shaped curve for the solidification velocity as a function of melt undercooling. The theory under consideration describes real experimental data on the solidification of Al-Ni melts.
{"title":"Mathematical Modeling of Measured Recalescence Velocities","authors":"Dmitri V. Alexandrov, Eugenya V. Makoveeva, Alexandra E. Glebova, Liubov V. Toropova","doi":"10.1002/mma.70204","DOIUrl":"https://doi.org/10.1002/mma.70204","url":null,"abstract":"<p>A moving-boundary problem of synchronous directional and volumetric solidification in undercooled liquid drops is formulated and solved. Directional solidification is caused by the temperature gradient along the spatial direction, whereas volumetric growth of new phase elements is caused by the melt undercooling ahead of the phase transition boundary. As this takes place, the solidification domain is divided into three regions occupied by solid material, two-phase layer, and liquid phase. The heat and mass transfer model is formulated inside all regions with the corresponding boundary conditions. To solve the problem, we use self-similar variables and functions, as well as the Laplace method for evaluating the Laplace-type integral and the technique of small parameter expansion of unknown functions in series. The temperature, impurity concentration, solid phase fraction distributions, and laws of motion of two solidification boundaries are found analytically. Our approximate solution shows that nucleation and growth of particles ahead of the solid phase: two - phase region boundary leads to a U-shaped curve for the solidification velocity as a function of melt undercooling. The theory under consideration describes real experimental data on the solidification of Al-Ni melts.</p>","PeriodicalId":49865,"journal":{"name":"Mathematical Methods in the Applied Sciences","volume":"49 2","pages":"977-988"},"PeriodicalIF":1.8,"publicationDate":"2025-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/mma.70204","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145772536","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Ulrich Abel, Ana-Maria Acu, Margareta Heilmann, Ioan Rasa
In this paper, we introduce a general family of Szász–Mirakjan–Durrmeyer type operators depending on an integer parameter . They can be viewed as a generalization of the Szász–Mirakjan–Durrmeyer operators, Phillips operators, and corresponding Kantorovich modifications of higher order. For , these operators possess the exceptional property to preserve constants and the monomial . It turns out that an extension of this family covers certain well-known operators studied before, so that the outcoming results could be unified. We present the complete asymptotic expansion for the sequence of these operators. All its coefficients are given in a concise form. In order to prove the expansions for the class of locally integrable functions of exponential growth on the positive half-axis, we derive a localization result, which is interesting in itself.
在本文中，我们引入了一组一般的Szász-Mirakjan-Durrmeyer类型算子，它们依赖于一个整数参数j∈k $$ jin mathbb{Z} $$。它们可以看作是Szász-Mirakjan-Durrmeyer算子、Phillips算子和相应的高阶Kantorovich修正的推广。对于j∈∈$$ jin mathbb{N} $$，这些运算符具有保留常数和单项x j $$ {x}&#x0005E;j $$的特殊性质。事实证明，这个家族的扩展涵盖了之前研究过的某些知名算子，从而可以统一结果。给出了这些算子序列的完全渐近展开式。它的所有系数都以简明的形式给出。为了证明一类指数增长的局部可积函数在正半轴上的展开式，我们得到了一个本身就很有趣的局部化结果。
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