On a novel approach for the investigation and approximation of solutions to the systems of higher order nonlinear PDEs

We study a boundary value problem for a system of the third order semi-linear partial differential equations with nonlocal boundary conditions. We establish sufficient conditions of existence, uniqueness, regularity and sign-preserving property of solutions of the studied problem and construct an iterative method for its approximation.


Introduction
Mathematical modeling of the processes of water filtration through the double-layered porous media [1], heat distribution in the heterogeneous environment [2], dampness distribution in the soil [11] lead to a scalar linear differential equation (DE) (1.2) u(t, x) + α(t, x)D (1.1) (1) where D (i, j) u(t, x) = ∂ i+ j u(t,x) ∂ i t∂ j x -denotes a mixed partial derivative of the function u(t, x) of the order i with resprect to t and of the order j with respect to x, m(t, x), α(t, x), d(t, x), η(t, x), a(t, x), b(t, x) and g(t, x) are given continuous functions in the domain of consideration.

u(t, x) + d(t, x)D (1.0) u(t, x) + η(t, x)D (0.2) u(t, x) + a(t, x)D (0.1) u(t, x) + b(t, x)u(t, x) = g(t, x),
Questions of existence and uniqueness of solutions to the mixed problems for the DE (1) under different local and nonlocal boundary conditions are studied in [5,13,14]. In [8,9] the authors investigate and construct approximate solutions to the boundary value problems (BVPs) in the case of systems of the third order semi-linear DEs under local and nonlocal boundary constraints. Authors also obtain sufficient conditions of existence and uniqueness of solutions to the studied BVPs, their signpreserving property and prove theorems about the differential inequalities.
The current paper is an extention of the results obtained in [8][9][10]. In particular, we study a BVP for a system of the third order semilinear partial differential equations (PDEs) coupled with the nonlocal boundary condition of the Nakhushev type. We construct a modification of the two-sided method to approximate a solution of the studied problem. In addition, we essentially improve the sufficient existence and uniqueness conditions for the solution, obtained earlier in [8,9].

Problem setting and auxiliary statements
Let us study the following problem: in the space of functions C * (D) where L 3 is a differential operator defined by the differential expression i j (t, x) , r = 1, 2, j = 1, n, are given matrices, δ i j is the Kronecker symbol, and the boundary conditions and T (x) := (τ i (x)), (t) := (ψ i (t)), (t) := (ω i (t)) are given vector-functions, From now on we assume that holds.
, then the BVP (2) and the system of integro-differential equations Here is a matrix.

Definition 1
We say, that a vector-function F[U (t, x)] ∈ C 3 (B), if it satisfies the following conditions: there exists a vector-function 3. vector-function H [U (t, x); V (t, x)] satisfies the Lipschitz condition, i.e. for arbi- x), r = 1, 2 and L is the Lipschitz matrix.

Remark 1 It is straightforward that if the vector-function F[U (t,
x)] ∈ C(B) and its first order partial derivatives with respect to all of its arguments starting from the third one are bounded, then ; -are functional matrices with non-negative coefficients satisfying the estimates: Let us construct sequences of vector-functions according to formulas: where for a zero approximation we take arbitrary in the space x) ∈ B 1 satisfying conditions: x) ∈ B satisfying conditions (11) are called the comparison functions to the BVP (2).
Note, that due to (9), (11) we have (10) we obtain: Taking into account inequalities (7), (9), (11), from (12)- (14) in virtue of the method of mathematical induction it is easy to check that if on every iteration step (10), (11) we pick components of the matrices C p,k 2 (t, x) and Q p,k 2 (t, x) such, that the conditions hold, then the constructed vector-functions , and in the domain B 1 there exist comparison functions Z 0 (t, x), V 0 (t, x) to the BVP (2), then the set of functional matrices C p,k 2 (t, x) and Q p,k 2 (t, x), satisfying conditions (15), is non-empty.
Proof Let us pick on every iteration step of (10), (11), (14) elements of the matrices Obviously, such non-negative functions c i, p,k 2 (t, x), q i, p,k 2 (t, x) satisfy conditions (9), and, due to (16), also the inequalities x) is a matrix, and The obtained inequalities prove the lemma.

Let us show that the constructed sequences of vector-functions {D
x)} uniformly converge to the same limit, that is a solution to the system of integro-differential equations (5). In virtue of (16) it is sufficient to show that From (13) follows that (19) From (19) using the mathematical induction method we obtain the estimates: Thus, From the estimates (20) it follows that i.e., It is easy to check that the limit vector-function U (t, x) is the solution to the integrodifferential system (5) and hence, to the BVP (2). (10), (11), (14) in the domain B 1 :

Theorem 2 Let conditions of the Theorem 1 to be hold. Then the sequences of vectorfunctions {Z p (t, x)}, {V p (t, x)} constructed by
1. uniformly converge to the unique regular solution of the BVP (2) for (t, x) ∈ D; 2. estimates (20) hold; 3. in the domain B 1 inequalities hold; 4. convergence of the method (10), (11), (14) is not slower than the convergence of the Picard method.

Proof
Let One can prove the uniqueness of solution to the BVP (2) and the inequality (21) by contradiction. For a detailed proof we refer to (Marynets et. al. 2019).
Let us prove statement 4 of the theorem. For this purpose assume, that Z p (t, x) and V p (t, x) are the comparison vector-functions of the problem (2). Then In virtue of the inequalities (7) and (9) Analogically we obtain that Hence, The last inequality finishes the proof.

Remark 2 1. Functions Z p (t, x) and V p (t, x) satisty the first two boundary conditions
in (3) and 2. Since for the p-th approximation to the exact solution we take the vector-functioñ thenŨ p (t, x) will satisfy all boundary conditions in (3). 3. It is worth mentioning that some approches for construction of the iterative methods with the improved convergence in the case of the operator equations were studied in [6,7]. Similar results for different classes of problems in the theory of differential equations were also obtained in [3,4,12].
then solution to the BVP (2) with the homogeneous boundary conditions (3) satisfies the inequalities: Together with the BVP (2) we consider the following problem: From now on we assume, that the right hand-sides of the problems (2) and (22) satisfy conditions below: (B), and in the domain B it has bounded first order partial derivatives with respect to Z (t, x) and 3. for an arbitrary vector-function where i, j (t, x) for some fixed D (0.k 2 ) Z (t, x) ∈ B, k 2 = 0, 1, and due to (24) It is straightforward that the vector-function satisfies the homogeneous boundary conditions (3) and H [W (t, x); 0] and Taking into account (26)-(28) and due to the Corollary 1 solution of the system (25) satisfies the inequalities: This completes the proof.

Example
Let us consider an illustrative example: in the space of functions C * (D 0 ), find a solution to a scalar differential equation coupled with the boundary conditions of the form: Note, that in the case of non-homogeneous boundary conditions they can always be reduced to the homogeneous ones.
The comparison characteristics of our computations are given in the Table 1.
From the results, presented in the table, follows that the convergence of the iterative method (10) in governed by C p,k 2 (t, x) and Q p,k 2 (t, x). Depending on their choice we can obtain different modifications to the considered method.
If necessary, one can continue the iteration process and construct further approximations to the exact solution with an even higher precision than those, obtained on the second iteration step.