The study of a mixed problem for one class of third order differential equations (original) (raw)
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2014
We prove two existence theorems for the integrodifferential equation of mixed type: x′(t) = f (t,x(t), ∫ t0k1(t,s)g(s,x(s))ds, ∫ a 0k2(t,s)h(s,x(s))ds), x(0) = x0, where in the first part of this paper f, g, h, x are functions with values in a Banach space E and inte-grals are taken in the sense of Henstock-Kurzweil (HK). In the second part f, g, h, x are weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functions f, g, h, x satisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness. Copyright © 2007 A. Sikorska-Nowak and G. Nowak. This is an open access article dis-tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop-erly cited. 1.
Nonlinear Integrodifferential Equations of Mixed Type in Banach Spaces
International Journal of Mathematics and Mathematical Sciences, 2007
We prove two existence theorems for the integrodifferential equation of mixed type:x'(t)=f(t,x(t),∫0tk1(t,s)g(s,x(s))ds,∫0ak2(t,s)h(s,x(s))ds),x(0)=x0, where in the first part of this paperf, g, h, xare functions with values in a Banach spaceEand integrals are taken in the sense of Henstock-Kurzweil (HK). In the second partf, g, h, xare weakly-weakly sequentially continuous functions and integrals are taken in the sense of Henstock-Kurzweil-Pettis (HKP) integral. Additionally, the functionsf, g, h, xsatisfy some conditions expressed in terms of the measure of noncompactness or the measure of weak noncompactness.
Theoretical and Numerical Discussion for the Mixed Integro– Differential Equations
Springer Nature, 2021
In this paper, we tend to apply the proposed modified Laplace Adomian decomposition method that is the coupling of Laplace transform and Adomian decomposition method. The modified Laplace Adomian decomposition method is applied to solve the Fredholm-Volterra integro-differential equations of the second kind in the space L2[a, b]. The nonlinear term will simply be handled with the help of Adomian polynomials. The Laplace decomposition technique is found to be fast and correct. Several examples are tested and also the results of the study are discussed. The obtained results expressly reveal the complete reliability, efficiency, and accuracy of the proposed algorithmic rule for solving the Fredholm-Volterra integro-differential equations and therefore will be extended to other problems of numerous nature.
Analytical discussion for the mixed integral equations
Springer International Publishing AG, part of Springer Nature 2018, 2018
This paper presents a numerical method for the solution of a Volterra–Fredholm integral equation in a Banach space. Banachs fixed point theorem is used to prove the existence and uniqueness of the solution. To find the numerical solution, the integral equation is reduced to a system of linear Fredholm integral equations, which is then solved numerically using the degenerate kernel method. Normality and continuity of the integral operator are also discussed. The numerical examples in Sect. 5 illustrate the applicability of the theoretical results.