On the Prandtl Equation

1999 ◽  
Vol 6 (6) ◽  
pp. 525-536
Author(s):  
R. Duduchava ◽  
D. Kapanadze
Keyword(s):  
Differential Equation ◽  
Sobolev Space ◽  
Unique Solvability ◽  
Bessel Potential ◽  
Integro Differential Equation ◽  
Prandtl Equation ◽  
Bessel Potential Spaces

Abstract The unique solvability of the airfoil (Prandtl) integro-differential equation on the semi-axis is proved in the Sobolev space and Bessel potential spaces under certain restrictions on 𝑝 and 𝑠.

scholarly journals Multidimensional Linear and Nonlinear Partial Integro-Differential Equation in Bessel Potential Spaces with Applications in Option Pricing

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1463
Author(s):  
Daniel Ševčovič ◽  
Cyril Izuchukwu Udeani
Keyword(s):  
Differential Equation ◽  
Option Pricing ◽  
Existence And Uniqueness ◽  
Trading Strategy ◽  
Shift Function ◽  
Bessel Potential ◽  
Stock Trading ◽  
Integro Differential Equation ◽  
The One ◽  
Bessel Potential Spaces

The purpose of this paper is to analyze solutions of a non-local nonlinear partial integro-differential equation (PIDE) in multidimensional spaces. Such class of PIDE often arises in financial modeling. We employ the theory of abstract semilinear parabolic equations in order to prove existence and uniqueness of solutions in the scale of Bessel potential spaces. We consider a wide class of Lévy measures satisfying suitable growth conditions near the origin and infinity. The novelty of the paper is the generalization of already known results in the one space dimension to the multidimensional case. We consider Black–Scholes models for option pricing on underlying assets following a Lévy stochastic process with jumps. As an application to option pricing in the one-dimensional space, we consider a general shift function arising from a nonlinear option pricing model taking into account a large trader stock-trading strategy. We prove existence and uniqueness of a solution to the nonlinear PIDE in which the shift function may depend on a prescribed large investor stock-trading strategy function.

scholarly journals Mixed type integro-differential equation with fractional order Caputo operators and spectral parameters

2021 ◽  
Vol 57 ◽  
pp. 190-205
Author(s):  
T.K. Yuldashev ◽  
E.T. Karimov
Keyword(s):  
Differential Equation ◽  
Fourier Series ◽  
Fractional Order ◽  
Unique Solvability ◽  
Mixed Type ◽  
Problem Solution ◽  
Algebraic Equations ◽  
Negative Part ◽  
Spectral Parameters ◽  
Integro Differential Equation

The issues of unique solvability of a boundary value problem for a mixed type integro-differential equation with two Caputo time-fractional operators and spectral parameters are considered. A mixed type integro-differential equation is a partial integro-differential equation of fractional order in both positive and negative parts of multidimensional rectangular domain under consideration. The fractional Caputo operator's order is less in the positive part of the domain, than the order of Caputo operator in the negative part of the domain. Using the method of Fourier series, two systems of countable systems of ordinary fractional integro-differential equations with degenerate kernels are obtained. Further, a method of degenerate kernels is used. To determine arbitrary integration constants, a system of algebraic equations is obtained. From this system, regular and irregular values of spectral parameters are calculated. The solution of the problem under consideration is obtained in the form of Fourier series. The unique solvability of the problem for regular values of spectral parameters is proved. To prove the convergence of Fourier series, the properties of the Mittag-Leffler function, Cauchy-Schwarz inequality and Bessel inequality are used. The continuous dependence of the problem solution on a small parameter for regular values of spectral parameters is also studied. The results are formulated as a theorem.

scholarly journals Asymptotic Solutions of Scalar Integro-Differential Equations with Partial Derivatives and with Fast Oscillating Coefficients

2020 ◽  
Vol 13 (2) ◽  
pp. 287-302
Author(s):  
Burkhan Kalimbetov ◽  
Akisher Temirbekov ◽  
Abdimuhan Tolep
Keyword(s):  
Differential Equation ◽  
Cauchy Problem ◽  
Differential Equations ◽  
Unique Solvability ◽  
Regularization Method ◽  
Singularly Perturbed ◽  
Integral Term ◽  
Integro Differential Equation ◽  
The Cauchy Problem ◽  
Oscillating Coefficients

In the paper, ideas of the Lomov regularization method are generalized to the Cauchy problem for a singularly perturbed partial integro-differential equation in the case when the integral term contains a rapidly varying kernel. Regularization of the problem is carried out, the normal and unique solvability of general iterative problems is proved.

Some numerical graphs with successive approximation method of fractional integro–differential equation

2019 ◽  
Vol 8 (4) ◽  
pp. 36
Author(s):  
Samir H. Abbas
Keyword(s):  
Differential Equation ◽  
Successive Approximation ◽  
Approximation Method ◽  
Existence And Uniqueness ◽  
Successive Approximation Method ◽  
Integro Differential Equation

This paper studies the existence and uniqueness solution of fractional integro-differential equation, by using some numerical graphs with successive approximation method of fractional integro –differential equation. The results of written new program in Mat-Lab show that the method is very interested and efficient. Also we extend the results of Butris [3].

Normalization Bernstein Basis For Solving Fractional Fredholm-Integro Differential Equation

2018 ◽  
pp. 491
Author(s):  
Abdul Khaleq O. Al-Jubory ◽  
Shaymaa Hussain Salih
Keyword(s):  
Differential Equation ◽  
Approximate Solution ◽  
Linear System ◽  
Differential Equations ◽  
Bernstein Basis ◽  
Traditional Methods ◽  
Integro Differential Equation

In this work, we employ a new normalization Bernstein basis for solving linear Freadholm of fractional integro-differential equations  nonhomogeneous  of the second type (LFFIDEs). We adopt Petrov-Galerkian method (PGM) to approximate solution of the (LFFIDEs) via normalization Bernstein basis that yields linear system. Some examples are given and their results are shown in tables and figures, the Petrov-Galerkian method (PGM) is very effective and convenient and overcome the difficulty of traditional methods. We solve this problem (LFFIDEs) by the assistance of Matlab10.   

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