{"title":"关于具有Caputo导数的分数阶迭代微分方程的解","authors":"Alemnew Abera, Benyam Mebrate","doi":"10.1155/2023/5598990","DOIUrl":null,"url":null,"abstract":"<jats:p>In this paper, we are concerned with two points. First, the existence and uniqueness of the iterative fractional differential equation <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\n <msup>\n <mrow />\n <mrow>\n <mi>c</mi>\n </mrow>\n </msup>\n <msup>\n <mrow>\n <mi>D</mi>\n </mrow>\n <mrow>\n <mi>α</mi>\n </mrow>\n </msup>\n <mi>c</mi>\n <mi>x</mi>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>t</mi>\n </mrow>\n </mfenced>\n <mo>=</mo>\n <mi>f</mi>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>t</mi>\n <mo>,</mo>\n <mi>x</mi>\n <mrow>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>t</mi>\n </mrow>\n </mfenced>\n </mrow>\n <mo>,</mo>\n <mi>x</mi>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>g</mi>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>x</mi>\n <mrow>\n <mfenced open=\"(\" close=\")\" separators=\"|\">\n <mrow>\n <mi>t</mi>\n </mrow>\n </mfenced>\n </mrow>\n </mrow>\n </mfenced>\n </mrow>\n </mfenced>\n </mrow>\n </mfenced>\n </math>\n </jats:inline-formula> are presented using the fixed-point theorem by imposing some conditions on <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\n <mi>f</mi>\n </math>\n </jats:inline-formula> and <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\n <mi>g</mi>\n </math>\n </jats:inline-formula>. Second, we proposed the iterative scheme that converges to the fixed point. The convergence of the iterative scheme is proved, and different iterative schemes are compared with the proposed iterative scheme. We prepared algorithms to implement the proposed iterative scheme. We have successfully applied the proposed iterative scheme to the given iterative differential equations by taking examples for different values of <jats:inline-formula>\n <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\n <mi>α</mi>\n </math>\n </jats:inline-formula>.</jats:p>","PeriodicalId":43667,"journal":{"name":"Muenster Journal of Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7000,"publicationDate":"2023-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"1","resultStr":"{\"title\":\"On Solutions to Fractional Iterative Differential Equations with Caputo Derivative\",\"authors\":\"Alemnew Abera, Benyam Mebrate\",\"doi\":\"10.1155/2023/5598990\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<jats:p>In this paper, we are concerned with two points. First, the existence and uniqueness of the iterative fractional differential equation <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M1\\\">\\n <msup>\\n <mrow />\\n <mrow>\\n <mi>c</mi>\\n </mrow>\\n </msup>\\n <msup>\\n <mrow>\\n <mi>D</mi>\\n </mrow>\\n <mrow>\\n <mi>α</mi>\\n </mrow>\\n </msup>\\n <mi>c</mi>\\n <mi>x</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>t</mi>\\n </mrow>\\n </mfenced>\\n <mo>=</mo>\\n <mi>f</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>t</mi>\\n <mo>,</mo>\\n <mi>x</mi>\\n <mrow>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>t</mi>\\n </mrow>\\n </mfenced>\\n </mrow>\\n <mo>,</mo>\\n <mi>x</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>g</mi>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>x</mi>\\n <mrow>\\n <mfenced open=\\\"(\\\" close=\\\")\\\" separators=\\\"|\\\">\\n <mrow>\\n <mi>t</mi>\\n </mrow>\\n </mfenced>\\n </mrow>\\n </mrow>\\n </mfenced>\\n </mrow>\\n </mfenced>\\n </mrow>\\n </mfenced>\\n </math>\\n </jats:inline-formula> are presented using the fixed-point theorem by imposing some conditions on <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M2\\\">\\n <mi>f</mi>\\n </math>\\n </jats:inline-formula> and <jats:inline-formula>\\n <math xmlns=\\\"http://www.w3.org/1998/Math/MathML\\\" id=\\\"M3\\\">\\n <mi>g</mi>\\n </math>\\n </jats:inline-formula>. Second, we proposed the iterative scheme that converges to the fixed point. The convergence of the iterative scheme is proved, and different iterative schemes are compared with the proposed iterative scheme. We prepared algorithms to implement the proposed iterative scheme. 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引用次数: 1
摘要
在本文中,我们关注两点。首先,迭代分数阶微分方程c D α的存在唯一性C x t = f t,xt,X g X t用不动点定理通过对f和g .其次,提出收敛于不动点的迭代方案。证明了该迭代方案的收敛性,并与所提出的迭代方案进行了比较。我们准备了算法来实现所提出的迭代方案。通过对不同α值的例子,我们成功地将所提出的迭代格式应用于给定的迭代微分方程。
On Solutions to Fractional Iterative Differential Equations with Caputo Derivative
In this paper, we are concerned with two points. First, the existence and uniqueness of the iterative fractional differential equation are presented using the fixed-point theorem by imposing some conditions on and . Second, we proposed the iterative scheme that converges to the fixed point. The convergence of the iterative scheme is proved, and different iterative schemes are compared with the proposed iterative scheme. We prepared algorithms to implement the proposed iterative scheme. We have successfully applied the proposed iterative scheme to the given iterative differential equations by taking examples for different values of .