scholarly journals Construction Solutions of Ordinary and Partial Differential Equations using the Analytical and Numerical Methods

2022 ◽  
Vol 3 (33) ◽  
pp. 103-120
Author(s):  
Turkia Dhawi Alqurashi ◽  
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Analytical Solution ◽  
Differential Equations ◽  
Finite Differences ◽  
Partial Differential ◽  
Internal Solution ◽  
Independent Variables ◽  
New Formula

In this paper we studied the solution of partial differential equations using numerical methods. The paper includes study of the solving partial differential equations of the type of parabolic, elliptic and hyperbolic, and the method of the net was used for the numerical nods, which represents a case of finite differences. We have two types of solution which are the internal solution and boundary solution. The internal solution is based on the internal nodes of the net. The boundary solution depends on the boundary nodes of the net, in addition to finding the analytical solution of the equations to compare the results. We also discussed solving the problem of Laplace, Poisson, for the importance of these equations in the applied side; Mat lab was used to find the values of tables for the values of border differences. We have derived a new formula for the solution of partial differential equations containing three independent variables.

A method for the numerical or mechanical solution of certain types of partial differential equations

1937 ◽  
Vol 161 (906) ◽  
pp. 353-366 ◽  
Keyword(s):  
Partial Differential Equations ◽  
Numerical Methods ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Formal Solution ◽  
Partial Differential ◽  
Numerical Work ◽  
Independent Variables ◽  
Wide Range ◽  
Technical Interest

The development of mechanical means of evaluating solutions of ordinary differential equations, in the form of the differential analyser of Dr. Bush (Bush 1931; Hartree 1935), has made it feasible to undertake the investigation of many problems of scientific and technical interest leading to differential equations which have no convenient formal solution, and which are too elaborate, or for which the range of solutions required is too extensive, for calculation of the solutions by numerical methods to be practicable. The practical success of this machine, and the wide range of equations to which it can be applied, have led to the hope that it may be found possible to apply it to partial differential equations, which are usually regarded as less amenable to numerical methods than ordinary equations. The present paper gives one way of applying it to such equations in two independent variables with certain types of boundary conditions. As will appear, the possibility of applying this method depends more on the form of the boundary conditions than on the exact form of the equations. The method is particularly suited to the differential analyser, though it is also practicable for numerical work.

Mixed Problems for Linear Systems of first Order Equations

1958 ◽  
Vol 10 ◽  
pp. 127-160 ◽  
Author(s):  
G. F. D. Duff
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Partial Differential ◽  
Auxiliary Data ◽  
First Order ◽  
Independent Variables ◽  
Mixed Problems ◽  
Linear Partial Differential Equations ◽  
Data Problem ◽  
Hyperbolic Differential Equations

A mixed problem in the theory of partial differential equations is an auxiliary data problem wherein conditions are assigned on two distinct surfaces having an intersection of lower dimension. Such problems have usually been formulated in connection with hyperbolic differential equations, with initial and boundary conditions prescribed. In this paper a study is made of the conditions appropriate to a system of R linear partial differential equations of first order, in R dependent and N independent variables.

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