Solving Initial Value Problem by Matching Asymptotic Expansions

2012 ◽  
Vol 72 (1) ◽  
pp. 405-416 ◽  
Author(s):  
Yuri Skrynnikov
Keyword(s):  
Asymptotic Expansions ◽  
Initial Value Problem ◽  
Initial Value

scholarly journals Singular Initial Value Problem for a System of Integro-Differential Equations

2012 ◽  
Vol 2012 ◽  
pp. 1-18 ◽  
Author(s):  
Zdeněk Šmarda ◽  
Yasir Khan
Keyword(s):  
Fixed Point ◽  
Asymptotic Behavior ◽  
Singular Point ◽  
Fixed Point Theorem ◽  
Asymptotic Expansions ◽  
Initial Value Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Initial Value ◽  
Analytical Properties ◽  
Singular Initial Value Problem

Analytical properties like existence, uniqueness, and asymptotic behavior of solutions are studied for the following singular initial value problem:gi(t)yi′(t)=aiyi(t)(1+fi(t,y(t),∫0+tKi(t,s,y(t),y(s))ds)),  yi(0+)=0,  t∈(0, t0], wherey=(y1, …, yn),  ai>0,  i=1, …, nare constants andt0>0. An approach which combines topological method of T. Ważewski and Schauder's fixed point theorem is used. Particular attention is paid to construction of asymptotic expansions of solutions for certain classes of systems of integrodifferential equations in a right-hand neighbourhood of a singular point.

scholarly journals On Inner Expansions for a Singularly Perturbed Cauchy Problem with Confluent Fuchsian Singularities

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 976
Author(s):  
Stephane Malek
Keyword(s):  
Cauchy Problem ◽  
Laplace Transform ◽  
Asymptotic Expansions ◽  
Initial Value Problem ◽  
Perturbation Parameter ◽  
Laplace Transforms ◽  
Singularly Perturbed ◽  
Initial Value ◽  
Logarithmic Kernel

A nonlinear singularly perturbed Cauchy problem with confluent Fuchsian singularities is examined. This problem involves coefficients with polynomial dependence in time. A similar initial value problem with logarithmic reliance in time has recently been investigated by the author, for which sets of holomorphic inner and outer solutions were built up and expressed as a Laplace transform with logarithmic kernel. Here, a family of holomorphic inner solutions are constructed by means of exponential transseries expansions containing infinitely many Laplace transforms with special kernel. Furthermore, asymptotic expansions of Gevrey type for these solutions relatively to the perturbation parameter are established.

scholarly journals Oscillation Criteria of Singular Initial-Value Problem for Second Order Nonlinear Dynamic Equation on Time Scales

2018 ◽  
Vol 5 (1) ◽  
pp. 102-112 ◽  
Author(s):  
Shekhar Singh Negi ◽  
Syed Abbas ◽  
Muslim Malik
Keyword(s):  
Time Scales ◽  
Initial Value Problem ◽  
Dynamic Equation ◽  
Nonlinear Dynamic ◽  
Type Inequality ◽  
Second Order ◽  
Oscillation Criterion ◽  
Initial Value ◽  
Nonlinear Dynamic Equation ◽  
Singular Initial Value Problem

AbstractBy using of generalized Opial’s type inequality on time scales, a new oscillation criterion is given for a singular initial-value problem of second-order dynamic equation on time scales. Some oscillatory results of its generalizations are also presented. Example with various time scales is given to illustrate the analytical findings.

scholarly journals A Neural Network Technique for the Derivation of Runge-Kutta Pairs Adjusted for Scalar Autonomous Problems

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1842
Author(s):  
Vladislav N. Kovalnogov ◽  
Ruslan V. Fedorov ◽  
Yuri A. Khakhalev ◽  
Theodore E. Simos ◽  
Charalampos Tsitouras
Keyword(s):  
Neural Network ◽  
Differential Evolution ◽  
Initial Value Problem ◽  
Fitness Function ◽  
Initial Value ◽  
Runge Kutta

We consider the scalar autonomous initial value problem as solved by an explicit Runge-Kutta pair of orders 6 and 5. We focus on an efficient family of such pairs, which were studied extensively in previous decades. This family comes with 5 coefficients that one is able to select arbitrarily. We set, as a fitness function, a certain measure, which is evaluated after running the pair in a couple of relevant problems. Thus, we may adjust the coefficients of the pair, minimizing this fitness function using the differential evolution technique. We conclude with a method (i.e. a Runge-Kutta pair) which outperforms other pairs of the same two orders in a variety of scalar autonomous problems.

scholarly journals On the sub–diffusion fractional initial value problem with time variable order

2021 ◽  
Vol 10 (1) ◽  
pp. 1301-1315
Author(s):  
Eduardo Cuesta ◽  
Mokhtar Kirane ◽  
Ahmed Alsaedi ◽  
Bashir Ahmad
Keyword(s):  
Asymptotic Behavior ◽  
Fractional Derivative ◽  
Initial Value Problem ◽  
Regularity Result ◽  
Time Variable ◽  
Variable Order ◽  
Initial Value ◽  
Integration By Parts ◽  
Integration By Parts Formula ◽  
Variable Order Fractional Derivative

Abstract We consider a fractional derivative with order varying in time. Then, we derive for it a Leibniz' inequality and an integration by parts formula. We also study an initial value problem with our time variable order fractional derivative and present a regularity result for it, and a study on the asymptotic behavior.

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